Mathematical and Computational Modelling of
Stochastic Partial Differential Equations Applied to
Advanced Materials
by
Inayatullah Soomro
A thesis submitted in partial fulfilment for the requirements for the degree of
Doctor of Philosophy at the University of Central Lancashire
September 2016
STUDENT DECLARATION FORM
I declare that while registered as a candidate for the research degree, I have not been a
registered candidate or enrolled student for another award of the University or other
academic or professional institution.
I declare that no material contained in the thesis has been used in any other submission
for an academic award and is solely my own work.
Signature of Candidate
Date: 20 September 2016
Type of award: Doctor of Philosophy
School: Physical Sciences and computing
I
Abstract
Soft materials have received increasing attention in the research community as
promising candidates due to their potential applications in the field of nanotechnology.
These applications include separation of membranes with nano-pores, templates for
nano-electronics, novel catalysts and drug delivery nano-particles. Block copolymers
have a fascinating property of self-assembly and are an important class of soft materials.
Confinement of the block copolymers is a very important tool for obtaining new
morphologies compared to bulk systems. In a confined system, interfacial interactions,
symmetry breaking, structural frustration, curvature and confinement-induced entropy
loss play vital roles in the formation of different morphologies compared to the bulk
systems. In this study classical lamella, cylinder and sphere forming morphologies were
investigated under geometric confinements. The Cell Dynamics Simulation (CDS)
method was used to study block copolymers under geometric confinements by
physically motivated discretizations. The CDS simulation method was implemented in
curvilinear coordinates systems to study accurately nanostructure formations of block
copolymer materials on curved surfaces. The CDS method provides a balance between
computational speed and physical accuracy. The novel nanostructures were obtained
with various film thicknesses, pore radii and with symmetric boundary conditions on the
film interfaces.
The formations of the lamella confined in the circular annular pore systems include:
concentric lamella, curved lamella, T-junctions, U-shaped, V-shaped, W-shaped, Y-
shaped and star shaped. Cylindrical and sphere forming systems confined in circular
annular pores showed novel packing arrangements along the spiral lines and in the
concentric circular rings. Meanwhile, a new set of CDS parameters with a high
temperature taken into consideration allowed study of the sphere forming system. The
confined lamellae in the cylindrical pores show novel nanostructures like helical
II
lamella, concentric lamella and perpendicular to the pore surface lamella. This study
reveals novel formation of tilted standing up cylinders packed into the concentric
circular rings confined in the cylindrical pores. The cylindrical forming system confined
in the cylindrical pores and spherical pores showed parallel and perpendicular cylinders;
curved cylinders, straight short cylinders, cylinders packed in concentric layers and
perforated holes. Sphere forming systems displayed various packing arrangements in
the cylindrical pores and the spherical pores geometries like hexagonal, pentagonal,
square, rhombus, concentric rings and along the vertices of the concentric quadrilateral.
The obtained novel nanostructures confined in the spherical pores are onion-like
nanostructures, stacked lamellar, perforated holes, standing lamella and coexistence of
lamella, cylinder and sphere morphologies.
III
Contents List of Figures .................................................................................................................... V
List of Tables ................................................................................................................... XIV
Acknowledgements ......................................................................................................... XV
Nomenclature ................................................................................................................ XVI
Greek Letters ................................................................................................................. XVII
CHAPTER ONE .................................................................................................................... 1
1. Introduction and aim of the investigation ................................................................. 1
1.1. Introduction and motivation of the study .......................................................... 1
1.2. Outline of thesis ................................................................................................. 2
1.3. Aims and objectives ............................................................................................ 4
CHAPTER TWO ................................................................................................................... 5
2. Block copolymers under geometric confinements: A literature review ................... 5
2.1. Overview ............................................................................................................. 5
2.2. Block copolymers in planar thin films .............................................................. 12
2.3. Block copolymers confined in cylindrical pores ............................................... 14
2.4. Block copolymers confined in spherical pores ................................................. 18
CHAPTER THREE .............................................................................................................. 21
3. Implementation of CDS method in polar, cylindrical and spherical coordinate
systems ............................................................................................................................ 21
3.1. Overview ........................................................................................................... 21
3.2. CDS and its implementation in polar, cylindrical and spherical coordinate
systems ........................................................................................................................ 24
3.3. Discretisation of Laplacian in the polar coordinate system ............................. 28
3.4. Discretisation of Laplacian in the cylindrical coordinate system ..................... 30
3.5. Discretisation of Laplacian in the spherical coordinate system ....................... 33
3.6. Simulations of binary fluid ................................................................................ 36
CHAPTER FOUR................................................................................................................ 38
4. Block copolymer system confined in circular annular pores ................................... 38
4.1. Introduction ...................................................................................................... 38
4.2. Results and discussions .................................................................................... 39
4.2.1. Asymmetric lamellae forming system confined in circular annular pores39
4.2.2. Symmetric lamella forming system confined in circular annular pores ... 55
4.2.3. Cylindrical forming system confined in circular annulus pores ................ 57
IV
4.2.4. Spherical forming system confined in circular annular pores .................. 62
4.3. Summary ........................................................................................................... 69
CHAPTER FIVE .................................................................................................................. 72
5. Block copolymers confined in cylindrical pores ...................................................... 72
5.1. Introduction ...................................................................................................... 72
5.2. Results and discussions .................................................................................... 73
5.2.1. Lamella forming system confined in cylindrical pores .............................. 73
5.2.2. Cylindrical forming system confined in cylindrical pores ......................... 81
5.2.3. Spherical forming system confined in cylindrical pores ........................... 92
5.3. Summary ......................................................................................................... 104
CHAPTER SIX .................................................................................................................. 106
6. Diblock copolymer system confined in spherical pores ........................................ 106
6.1. Introduction .................................................................................................... 106
6.2. Results and discussion .................................................................................... 107
6.2.1. Lamella forming system confined in spherical pores ............................. 107
6.2.2. Cylindrical forming system confined in spherical pores ......................... 116
6.2.3. Sphere forming system confined in spherical pores ............................... 124
6.3. Summary ......................................................................................................... 132
CHAPTER SEVEN ............................................................................................................ 134
7. Conclusions and Future work ................................................................................ 134
7.1. Block copolymers confined in the circular annular pores .............................. 134
7.2. Block copolymers confined in cylindrical pores ............................................. 135
7.3. Block copolymers confined in spherical pores ............................................... 138
7.4. Future work .................................................................................................... 139
References ..................................................................................................................... 140
V
List of Figures
Figure 2.1: The bulk phase diagram of diblock copolymer system [25]. The horizontal
axis shows volume fraction of A segment of polymer system, vertical axis on the left
illustrates Flory-huggins interaction parameter and vertical axis on the right presents the
decreasing temperature parameter. In the diagram (LAM) indicates lamella phase,
(HEX) indicates hexagonal packed cylindrical phase, (BCC) is for body-centred-cubic
spheres phase and (GYR) refers to the bi-continues complex gyroid phase in the bulk
system. ............................................................................................................................... 8
Figure 2.2: AB diBlock copolymers nanostructures in the bulk systems [22]................ 10
Figure 2.3: Schematic diagrams of different architectures of block copolymers. The
green represents the block A, red one represents the block B and the blue one represents
block C. (a) A-B diblock copolymer, (b) A-B-A triblock copolymer, (c) Grafted or
branched A-B diblock copolymer, (d) Star A-B-C triblock copolymer, (e) Linear A-B-C
triblock copolymer. ......................................................................................................... 11
Figure 3.1: Polar mesh diagram. ..................................................................................... 28
Figure 3.2: Cylindrical mesh diagram [82]. .................................................................... 30
Figure 3.3: Sphere grid system diagram [83] .................................................................. 33
Figure 3.4: Phase separation for binary fluid were obtained on 1 million time steps, in a
pore system with the interior radius of the pore system was fixed at ar = 3.0 and the grid
sizes are (a) 30x360 (b) 60x360 ...................................................................................... 37
Figure 4.1: Evolutions of asymmetric lamellae system on 1 million time steps confined
in various circular annular pore sizes with the interior radius of the pore was fixed at
0.3ar and the exterior radius of pore was expanded to obtain various pore sizes: (a)
evolution of asymmetric lamella in the pore system size d = 1, pore system induces
alternating lamella strips (b) evolution of asymmetric lamella in the pore system size d
= 2, system patterned into Y-shaped, U-shaped and perforated holes morphologies (c)
VI
evolution of asymmetric lamella in the pore system size d = 3, pore shows Y-shaped or
tilted Y-shaped star shaped and W-shaped patterns (d) evolution of asymmetric lamella
in the pore system size d = 4, system induces Y-shaped, W-shaped U-shaped and a few
perforated holes (e) evolution of asymmetric lamella in the pore system size d = 5,
patterns are two arm stars and single arm star with a perforated hole at the centre of the
star (f) evolution of asymmetric lamella in the pore system size d = 6, induced
nanostructures include perforated holes morphology (g) evolution of asymmetric
lamella in the pore system size d = 8, mixed morphologies (h) evolution of asymmetric
lamella in the pore system size d = 10, alternating lamella strips normal to the exterior
circular boundary. ........................................................................................................... 41
Figure 4.2: Evolution of asymmetric lamellae system on 1 million time steps in the pore
system with the interior radius of pore system was fixed at ar = 5.0 and the exterior
radius br of the pore system was expanded to obtain various pore sizes, pore size of the
systems: (a) evolution of nanodomains in the pore size d = 2, system show alternating
lamella strips normal to the circular boundaries, perforated holes and Y-shaped defects
(b) evolution of lamella patterns in the pore system size d = 3 system induces lamella
strips and Y-shaped defects (c) evolution of the asymmetric lamella system in the pore
size d = 4 induced morphologies are Y-shape, U-shape, W-shape and star shape
lamellae with a perforated hole at the centre of the star (d) nanostructure evolution in
the pore size d = 5 induces diverse lamella patterns. ...................................................... 45
Figure 4.3: The asymmetric lamellae system in the expanded pore size with the interior
radius of pore system was fixed at ar = 7: (a) the system size d = 2, system induces
alternating parallel lamella strips normal to the circular boundaries, Y-shaped, W-
shaped and star shaped patterns (b) the system size d = 3, induced patterns including a
few alternating lamella strips normal to the circular boundaries, Y-shaped, W-shaped
and star shaped defects. ................................................................................................... 46
VII
Figure 4.4: Asymmetric lamellae system under one dimensional confinement obtained
on 1 million time steps, the interior radius of the pore system was fixed at ar =3.0 and
the pore sizes of systems are (a) d = 1, pore system induces three alternating concentric
lamella rings (b) d = 2.5, pore system induces seven alternating concentric lamella rings
(c) d = 3, system induced concentric lamella and strip lamella (d) d = 4, system induced
concentric lamella and dislocations (e) d = 6, system patterned with mixed lamella
morphologies. .................................................................................................................. 48
Figure 4.5: Evolution of asymmetric lamellae system on 1 million time steps under
geometric confinement, interacting circular walls has affinity with the B segment of
polymer system, interaction strength was applied 2.0 , the interior radius of the pore
system was fixed at ar = 3, whereas system was expanded by the exterior radius of the
pore system and pore system size are (a) d = 1, system patterned into concentric lamella
(b) d = 2.5, system induces concentric lamella with very few dislocations (c) d = 3,
concentric lamella, perforated holes and dislocations (d) d = 4, mixed patterns............ 49
Figure 4.6: Asymmetric lamellae system under one dimensional confinement into
expanded pore system with the interior radius was fixed at ar = 5 and the pore size of
the systems are (a) d = 2.5 (b) d = 3 (c) d = 4. ............................................................... 51
Figure 4.7: Asymmetric lamellae system under geometric confinement, circular walls
have affinity to the B segment of the polymer system, the interior radius of the pore
systems was fixed at ar = 5, and the pore sizes are (a) d = 2 (b) d = 2.5 (c) d = 3 (d) d =
4. ...................................................................................................................................... 52
Figure 4.8: Asymmetric lamellae system under geometric confinement, the interaction
strength was applied 2.0 , the expanded pore system with the interior radius of the
pores were fixed at ar = 7 and pore sizes of the systems are (a) d = 2.5 (b) d = 3.5. ..... 53
Figure 4.9: Lamellae system under geometric confinement, circular walls have affinity
to the B segment of the polymer system, the interior radius of the pore system was fixed
VIII
at ar = 7, the pore size expanded by the exterior radius of the pore system, pore sizes are
(a) d = 2.5 (b) d = 3.5. .................................................................................................... 54
Figure 4.10: Symmetric lamellae forming system confined in circular annular pores with
the interior radius of the pore system was fixed at ar = 3.0 and the pore sizes are (a) 2 (b)
4 (c) 6. ............................................................................................................................. 56
Figure 4.11: Symmetric lamellae system under geometric confinement with the interior
radius of the pore system was fixed at ar = 3.0 and pore sizes of the system are (a) d = 3
(b) d = 6. ......................................................................................................................... 57
Figure 4.12: Cylindrical system in the pore geometry with pore sizes (a) d = 1 (b) d = 2
(c) d = 3 (e) d = 4 (f) d = 6 ............................................................................................. 58
Figure 4.13: Cylindrical systems confined under interfacial circular walls having
affinity to A block and the pore size of systems are (a) d = 2 (b) d = 3 (c) d = 4 (d) d =
6. ...................................................................................................................................... 60
Figure 4.14: Cylindrical systems with interfacial affinity to B block and the pore size of
the systems are (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.0. ............................................... 61
Figure 4.15: Sphere system in the circular annular pore geometry and the pore sizes are
(a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6. ............................................................................ 63
Figure 4.16: Sphere forming system with modified CDS parameter system and pore
system sizes are (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6. ................................................. 64
Figure 4.17: sphere forming system with system subject to the affinity of interacting
walls to A block and system size (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6. ...................... 65
Figure 4.18: Sphere system with preferential attractive wall for monomer B and the pore
system size are (a) d = 2 (b) d = 3. ................................................................................. 66
Figure 4.19: Sphere forming system with modified CDS parameter system subject to the
affinity of interacting walls to A block and system sizes are (a) d = 2 (b) d = 3 (c) d = 4
(d) d = 6. ......................................................................................................................... 67
IX
Figure 4.20: Sphere forming system with modified CDS parameter system subject to the
affinity of interacting walls to A block and system size (a) d = 2 (b) d = 3 (c) d = 4 (d)
d = 6. ............................................................................................................................... 68
Figure 5.1: Lamella system confined in neutral cylindrical pores, with pore system
where interior radius was fixed at ar = 3 and the pore size (a) d = 0.5 (b) d = 1.5 (c) d =
2. ...................................................................................................................................... 74
Figure 5.2: Lamellae system confined in the neutral cylindrical pores, interior radius of
pore system fixed at ar = 5 and pore size (a) d = 0.5 (b) pore size d = 1 (c) d = 1.5 (d) d
= 2 (e) d = 2.5 (f) d = 3. ................................................................................................. 75
Figure 5.3: Lamella system with interfacial surfaces, the pore system with interior
radius fixed at ar = 3 and the pore sizes are (a) d = 1.0 (b) d = 1.5. .............................. 76
Figure 5.4: Lamella system with interfacial surfaces, interior radius of the pore surface
was fixed at 5ar and the pore sizes are (a) d = 1.0 (b) d = 1.5(c) d = 2.5 (d) d = 3.0. 77
Figure 5.5: Cylindrical system confined by interfacial surfaces perpendicular to the pore
system and the pore size and pore lengths are (a) d = 1.6 and h = 1.6 (b) d = 2.5 and h
= 1.0 (c) d = 2.5 and h = 2.5. ......................................................................................... 79
Figure 5.6: Two dimensional confinement of cylindrical system with the pore sizes and
the pore lengths are (a) d = 1.0 and h = 1 (b) d = 1.0 and h = 1.6. ................................ 81
Figure 5.7: Cylindrical forming system confined in neutral surfaces, in a pore system
with the interior radius fixed at ar = 3, the pore sizes of the system are (a) d = 1 (b) d =
1.5 (c) d = 3. .................................................................................................................... 82
Figure 5.8: Cylindrical morphology in the neutral cylindrical pores with a pore system
whose the interior radius was fixed at 5ar and the pore size of systems are (a) d = 0.5
(b) d = 1.0 (c) d = 1.5. .................................................................................................... 84
X
Figure 5.9: Cylinder system under interfacial surfaces in a pore system with the interior
radius fixed at 3ar and the pore sizes of system are (a) d = 1 (b) d = 1.5 (c) d = 2 (d) d
= 2.5 (e) d = 3. ................................................................................................................ 85
Figure 5.10: Cylindrical forming system under surface interacting parameter 2 , the
interior radius of pore system is fixed at 5ar , the length of pore was fixed at h = 6,
20 and the pore sizes are (a) d = 0.9 (b) d= 1.0 (c) d = 1.5 with the interior layer
(d) d = 2 (e) d = 2.5 (f) d = 3.......................................................................................... 86
Figure 5.11: Cylindrical forming system with fixed pore size d =5, whereas, the height
was varied for different values of lattice. ........................................................................ 89
Figure 5.12: cylindrical system obtained with interaction parameter 4.0 , the pore size
and the pore lengths respectively are (a) d = 1.6 and h = 1.6 (b) d = 1.5 and h = 4 (c) d =
2 and h=4. ........................................................................................................................ 91
Figure 5.13: Sphere forming system in neutral cylindrical pore systems with low
temperature CDS parameter system, the interior radius of pore system was fixed at ar =
3 and the pore sizes are (a) d =2.5 (b) d = 3.0. ............................................................... 93
Figure 5.14: Sphere system obtained by low temperature CDS parameter system, the
pore system has the interior radius fixed at ar =5 and pore sizes in the images are (a) d =
0.5 (b) d = 1 (c) d = 1.5. ................................................................................................. 93
Figure 5.15: Sphere system obtained by modified CDS parameters system with high
temperature in a pore system with the interior radius fixed at ar = 3 and the pore sizes
are (a) d = 1 (b) d = 2 (c) d = 2.5. .................................................................................. 95
Figure 5.16: Sphere system obtained with modified CDS parameter system in a pore
system whose interior radius were fixed at ar = 0.5and the pore sizes of the systems are
(a) d = 0.5 (b) d = 1 (c) d = 1.5 (d) d = 2 (e) d = 2.5 (f) d = 3. ..................................... 96
Figure 5.17: Sphere system with interacting parallel circular walls having preferential
affinity with majority segment of the polymer system, in a pore system with interior
XI
radius of the pore were fixed at ar = 3 and the pore sizes of the system are (a) d = 1.5
(b) d = 2 (c) d = 2.5. ....................................................................................................... 98
Figure 5.18: Sphere system under interfacial cylindrical surfaces and size of pore
systems are (a) d = 1.5 (b) d = 2 (c) d = 2.5 (d) d = 0.3. ............................................... 99
Figure 5.19: Sphere system under geometric confinement applied through cross sections
of cylindrical pore system with the pore size and the pore length are respectively (a) d =
1.5, h = 1.2 (b) d = 2.5, h = 2.5 (c) d = 3.0, h = 3.0. ................................................... 101
Figure 5.20: Sphere system under two dimensional confinements with system size and
length are respectively (a) d = h = 1.5 (b) d = h = 2.5 (c) d = h = 3.0. ....................... 103
Figure 6.1: Asymmetric lamellae system in spherical pores with neutral surfaces, the
interior radius of the pores were fixed at 3ar and the pore sizes of the systems are (a) d
= 0.5 (b) d = 1 (c) d = 1.5 (d) d = 3. ............................................................................ 108
Figure 6.2: Asymmetric lamella system in the pore surface with the interior radius fixed
at 5ar and shell sizes are (a) d = 0.5 (b) d = 1 (c) d = 1.5 (d) d = 3. ........................ 110
Figure 6.3: Asymmetric lamella system in spherical pore with the interior radius of the
pore system fixed at 7ar and the spherical pore sizes are (a) d = 0.5 (b) d = 1 (c) d =
1.5 (d) d = 3. ................................................................................................................. 111
Figure 6.4: Lamella systems with interfacial surfaces, the pore surface having the
interior radius fixed at 3ar and the pore sizes are (a) d = 1 (b) d = 1.5. ..................... 112
Figure 6.5: Lamella systems confined by interfacial interfaces, with pore surface having
the interior radius fixed at 5ar and the pore sizes are (a) d = 1 (b) d = 1.5. ............... 113
Figure 6.6: lamella systems confined by interacting interfaces, with the pore surface
having the interior radius fixed at 7ar and the pore sizes are (a) d = 1 (b) d = 1.5 (c) d
= 2.0 (d) d = 2.5 (e) d = 3.0.......................................................................................... 114
XII
Figure 6.7: Cylindrical forming system confined in neutral spherical pores with the
interior radius of the pore surface fixed at 5ar and the pore sizes are (a) d = 0.5 (b) d =
1 (c) d = 2.5 (d) d = 3.0. ............................................................................................... 117
Figure 6.8: Cylindrical forming system confined in neutral spherical pores with the
interior radius of the pore surface fixed at 7ar and the pore sizes are (a) d = 0.5 (b) d =
2.0 (c) d = 2.5 (d) d = 3.0. ............................................................................................ 119
Figure 6.9: Cylinder system confined by parallel spherical interfaces in a spherical pore
with the interior radius fixed at 5ar and the pore surface sizes are (a) d = 1 (b) d = 1.5
(c) d = 2.5. ..................................................................................................................... 121
Figure 6.10: Cylinder system confined by parallel spherical interfaces in a spherical
pore with the interior radius fixed at 7ar and the pore surface sizes are (a) d = 1 (b) d
= 1.5 (c) d = 2.0. ........................................................................................................... 123
Figure 6.11: Modified sphere system in the neutral pore surfaces with the interior radius
of the pore surface fixed at 3ar and the pore sizes are (a) d = 0.5 (b) d = 1.0. .......... 125
Figure 6.12: Modified sphere system in the neutral pore surfaces with the interior radius
of the pore surface fixed at 5ar and the pore sizes are (a) d = 0.5 (b) d = 1.0 (c) d =
1.5. ................................................................................................................................. 126
Figure 6.13: Sphere system obtained by modified CDS parameter system in the neutral
pore surfaces with the interior radius of the pore surface fixed at 7ar and the pore sizes
are (a) d = 0.5 (b) d = 1.0 (c) d = 1.5. .......................................................................... 127
Figure 6.14: Sphere system confined by parallel to the pore walls, with the interior
radius of the pore surface fixed at 5ar and the pore surface sizes are (a) d = 1.5 (b) d =
2.0 (c) d = 2.5. ............................................................................................................... 129
XIII
Figure 6.15: Sphere system confined by parallel to the pore walls, with the interior
radius of the pore surface fixed at 7ar and the pore surface sizes are (a) d = 1.5 (b) d =
2.0 (c) d = 2.5. ............................................................................................................... 131
XIV
List of Tables
Table 1: Computational parameters used in the simulation results ................................ 39
Table 2: CDS Parameters for asymmetric lamella forming system ................................ 40
Table 3: CDS parameters for symmetric lamella system ................................................ 55
Table 4 Modified CDS parameters for sphere forming system ...................................... 64
Table 5: CDS parameter system for asymmetric lamella forming system...................... 74
Table 6: CDS parameter system with low temperature for sphere forming system ....... 92
Table 7: Modified CDS parameter system for sphere forming system ........................... 95
Table 8: CDS parameter system for sphere forming system for low temperature .......... 98
Table 9: CDS parameter system for asymmetric lamella forming system.................... 107
Table 10: CDS parameter system for cylindrical forming system ................................ 116
Table 11: Modified CDS parameter system for sphere forming system with high
temperature .................................................................................................................... 124
Table 12: CDS parameters for sphere forming system with low temperature .............. 128
XV
Acknowledgements
I am grateful to all the people who provided support, encouragement, advice and
friendship during my studies at the University of Central Lancashire. It is a great
pleasure to acknowledge all of them here.
I would like to first express my sincere gratitude to my supervisors, Dr Dung Ly and
Prof Dr Waqar Ahmed for their vital guidance, patience and encouragements during the
study. I am thankful for your confidence in me and your eagerness.
I also would like to thank you Prof Dr Andrei Zvelindovsky and Dr Marco Pinna for
their valuable support.
I would like to thank you, my colleagues and friends of University of Central
Lancashire and Preston city for their valuable motivations and help.
I wish to acknowledge the Shah Abdul Latif University Khairpur, Sindh, Pakistan and
Higher Education Commission (HEC) Pakistan for funding and support during the
study.
In the last but not the least, enormous and special thanks to my family members for their
support and love. I am especially grateful to my wife; in my absence she provided care
and love to my kids. This journey would not have been possible without the support of
my wife, my mother and rest of family members. I dedicate this thesis to all my family
members.
XVI
Nomenclature
A-----------------------------------------------------------------------Phenomenological constant
B ---------------------------------------------------Chain-length dependence to the free energy
A-----------------------------------------------------------------------Phenomenological constant
B-----------------------------------------------------------------------Phenomenological constant
D -----------------------------------------------------Positive constant for diffusion coefficient
F ---------------------------------------------------------------------------Free energy functional
Af -----------------Global volume fraction of A monomers in the diblock polymer system
f -----------------------------------------------------------------------------------Map function
F -----------------------------------------------------------------------Free energy functional
H -------------------------------------------------------------------------Free energy function
M ----------------------------------------------------------Phenomenological mobility constant
AN ------------------------------------------------------------------------Number of A monomers
BN ------------------------------------------------------------------------Number of B monomers
rN ---------------------------------The length of the radial r dimension of the lattice system
N ---------------------------------The length of the angular dimension of the lattice system
zN ----------------------------------------The length of the z dimension of the lattice system
N -----------------------------The length of the azimuthal angular dimension of the lattice
t ---------------------------------------------------------------Time evolution of order parameter
u ----------------------------------------------------------------------Phenomenological constant
v -----------------------------------------------------------------------Phenomenological constant
XVII
Greek Letters
-----------------------------------------------------------------------------------Order parameter
2 -----------------------------------------------------------------------The Laplacian operator
--------------------------------------------------The averaging of the stencil in the lattice
A --------------------------------------------------------Local volume fraction for A monomer
B --------------------------------------------------------Local volume fraction for B monomer
-The interaction strength between pore interfaces and one of the segments of the
polymer system
----------------------------------------------------------------------Temperature-like parameter
t------------------------------------------------------------------------------------------Time-step
r -------------------------------------------------------------------------Radial step in the lattice
------------------------------------------------------------------The angular step in the lattice
z -----------------------------------------------------The length steps in the cylindrical lattice
------------------------------------------The azimuthal angular step in the spherical lattice
--------weighting factor of nearest neighbour points for laplacian in polar coordinates
---weighting factor of nearest neighbour points for laplacian in cylindrical coordinates
-- weighting factor of nearest neighbour points for laplacian in spherical coordinates
1
CHAPTER ONE
1. Introduction and aim of the investigation
1.1. Introduction and motivation of the study
Block copolymers due to their ability to self-assemble into rich nanostructures and
potential scientific applications have attracted tremendous scientific interest. Different
numbers of ordered equilibrium phases like lamellae, hexagonally packed cylinders,
body centred-cubic spheres, and a bi-continues network structure gyroid phases have
been identified in diblock copolymer melts. Nanostructure control is a prime goal for
researchers and can be achieved by applying external fields, particularly surface fields
which are prominent in the confining geometries. Under geometric confinements, more
complex structures, different morphologies and symmetry breaking of an ordered
structure can be achieved.
In this contribution we focus on mathematical modelling and computer simulations of
diblock copolymers (two blocks per molecule) in various confinements. Due to the very
large parameter space experimental study of block copolymers is very time consuming.
However, computer simulations can help in finding optimal parameters to create new
block copolymer materials. Most of the previous computational studies were carried out
for block copolymers under geometric confinements by using the Cartesian coordinate
system. While, block copolymers confinements in circular, spherical and cylindrical
pores using the Cartesian coordinate system could lead to artefacts in the systems. In
order to study accurately block copolymer confined in circular, cylindrical and spherical
pores the CDS method was employed in polar, cylindrical and spherical coordinate
systems. In order to account for curvature effects at each point on the lattice, isotropic
2
discrete Laplacian operators were derived in polar, cylindrical and spherical coordinate
systems. Therefore, this study provides the physical motivated discretization of curved
surfaces. A mathematical model for the order parameter which involves partial
differential equations was solved by numerical approximations using finite difference
schemes. The CDS equations were solved with the appropriate boundary conditions on
the circular thin film, the cylindrical pores and the spherical pore surfaces. The CDS
simulation codes were developed in the FORTRAN computer language for the
evaluations of the order parameter in polar, cylindrical and spherical coordinate
systems.
Further, in this study, confinement-induced nanostructure formations have been carried
out in the natural geometric environment by using polar, cylindrical and spherical
coordinate systems. The curvilinear coordinate systems provide accurate nanostructure
formations in the confined systems as compared to the Cartesian coordinate system. In
the case of confinement, the effect of interfacial surfaces can also be accurately studied
by using polar, cylindrical and spherical coordinate system. The study reveals various
novel nanostructures confined in circular annular pores, cylindrical pores and spherical
pores with various pore sizes and pore radii.
1.2. Outline of thesis
In this study investigations were carried out on self-assembly of diblock copolymers
confined in annular circular pores, hollow cylindrical pores, and spherical nanopores.
To study accurately, block copolymers confined in circular annular pores, cylindrical
pores, and spherical pores, the CDS method was employed in polar, cylindrical and
spherical coordinate systems.
3
In Chapter 2 a literature review of block copolymers confined in the planar thin films,
the cylindrical pores and the spherical pore is given.
In chapter 3 the mathematical model and algorithm are discussed in detail. Evaluation
of order parameter by partial differential equations and its allied equations which
represent the CDS method are discussed in detail. The discretized isotropic forms of
Laplacian operators in polar, cylindrical and spherical coordinate systems were derived
and discussed.
In chapter 4 block copolymers are investigated under geometric confinements of the
circular annular pores with various film thicknesses and pore radii. Lamella, cylinder,
and sphere forming systems are studied in the circular annular pore system with neutral
surfaces and interfacial surfaces. Results predicted novel nanostructure in the circular
annular pore geometry. Studies are carried out with interacting circular walls having
affinity with the majority block as well as the minority block of the polymer system. A
new set of CDS parameters for sphere forming system is proposed.
In chapter 5 studies are carried out on block copolymers confined in the hollow
cylindrical pore with various sizes of the pore surfaces, pore lengths and pore radii.
Lamella, cylinder, and spherical forming systems are obtained and discussed in the
cylindrical pore geometry. The results are obtained and discussed with the neutral pore
surfaces and the interfacial pore surfaces having affinity to the majority segment of the
polymer system. Block copolymers confined in the cylindrical pores are obtained and
discussed with one-dimensional confinements and two-dimensional confinements.
In chapter 6 studies and investigations are carried out for block copolymers confined in
the spherical shell with various thicknesses of the pore surfaces and pore radii. Lamella,
cylinder, and sphere systems are obtained and discussed in the spherical pore geometry.
4
Results are obtained and discussed with neutral pore surfaces as well as interfacial
surfaces having affinity to the majority segment of the polymer system.
In chapter 7 covers conclusions, discussions, and future works of this study.
1.3. Aims and objectives
Following are the aims and objectives of this study.
I. To study and investigate self-assembled diblock copolymers lamella, cylindrical
and spherical forming systems confined in annular circular pores by using the
CDS method employed in the polar coordinate system.
II. To investigate self-organizing diblock copolymers lamella, cylindrical and
spherical forming nanostructures confined in hollow cylindrical pores by using
the CDS method employed in the cylindrical coordinate system.
III. To Study and analyses of self-assembling diblock copolymers lamella,
cylindrical and spherical forming systems confined in spherical pores by using
the CDS method employed in the spherical coordinate system.
5
CHAPTER TWO
2. Block copolymers under geometric confinements: A
literature review
2.1. Overview
The diblock copolymer system is an important class of soft materials which are
characterised by a self-assembling property [1, 2]. The self-assembling property enables
diblock copolymers for potential scientific applications. [3]. Geometric confinements of
block copolymers provide spatial constraints to the molecules of the polymer system.
Nano structural control is a prime goal for researchers. Confinement is a significant tool
to break the symmetry of an ordered structure [4]. Under the constraint environment, the
block copolymers conforms novel nanostructures which is important for significant
biological and chemical applications [5]. Block copolymers are one of the classes of soft
materials which consist of different polymer blocks covalently connected to each other
and the length scale of the ordered structures at nanometer scale is (10-100 nm) [6-8].
AB diblock copolymers are the simplest linear polymer systems which consist of two
blocks interconnected covalently to each other [9, 10]. Due to the fact that the phase
separation is driven by chemical incompatibilities between the different blocks, they can
spontaneously form various patterns of nanostructures depending on block-block
interactions, block composition, architecture of blocks, molecular characteristics and
furthermore, external conditions such as the geometry of confinement [3, 9, 10]. A
number of ordered microdomain structures have been explored in diblock copolymer
melts such as lamellae, hexagonally packed cylinders, body-centred arrays of spheres
and complex structures such as bi-continuous network structures [7, 10-14]. More
6
complex block copolymers and various morphologies can be obtained by placing block
copolymers under geometrical confinements [6, 10]. Block copolymers under
geometrical confinement, conform many fascinating nanostructures that do not tend to
conform in the bulk systems. Spatial confinement provides a powerful tool to break the
symmetry of an ordered nanostructure [15] and paves the way for fabricating novel
morphologies [8]. Under confinement, interfacial interactions, symmetry breaking,
structural frustrations and confinement-induced entropy can play vital roles in the
nanostructural control of soft materials [10, 16]. The pore is the most reasonable
geometry to be considered and its effects on morphology formation are significant for
nanotechnology applications and biology [17]. The curvature and interfacial interactions
influence microdomains to change their orientation [7]. To avoid the difficulty
associated with irregular boundaries or arbitrarily shaped boundaries, the entire physical
problem can be transformed into boundary-fitted curvilinear coordinate systems such as
polar, spherical and cylindrical coordinates [18]. Cylindrical geometry provides a high
degree of curvature imposed on the surface of nanopores which causes a frustration of
chain packing at the interface, leading to the formation of new morphologies [15].
Previous studies were done on one-dimensional (thin film) and two-dimensional
cylindrical pores confined systems and under three-dimensional confinements (spherical
and elliptical) pores [19]. To the best of our knowledge, no one has studied diblock
copolymers under geometric confinement using the CDS method applied in polar and
cylindrical coordinates. However, it is our aim and desire to fill this gap by providing a
complete understanding of applying the CDS method using polar, cylindrical and
spherical coordinate systems in the investigation of the soft materials.
Previous studies of block copolymers were carried out by using a square lattice, but
unfortunately, these studies did not provide full isotropic results while studying two-
dimensional confinements with polar coordinates may give full isotropic results as
7
being the nature of geometrical coordinates. These confinements may give
generalizations and rules about controlling the morphologies as a geometric function.
Using CDS, the morphology of thin block copolymers films around a spherical
nanoparticle has been investigated to obtain nanoshells with structural control through
the interplay between film thicknesses, characteristics of block copolymers, surface
parameters, and curvature [20]. Block copolymers confined in cylindrical nanopores
have been studied systematically and have found novel structures such as helices and
stacked toroids inside the cylindrical pores. These types of confinement-confinement
induced morphologies depend on the pore diameter and the surface-polymer
interactions, which shows the importance of structural frustration as well as interfacial
interactions [14]. Later the pore diameter and surface-polymer interactions were varied
systematically to investigate their influence on the self-assembled morphologies and
chain conformations. In the case of Lamella-forming and cylinder-forming, block
copolymers under cylindrical pore confinement novel structures were reported, such as
helices and concentric lamellae [21]. Different structures including stacked-disk, single-
helix, gyroidal, catenoid-cylinder, stacked-circle and concentric cylindrical barrel
morphologies have been obtained depending on the diameter of the confining
cylindrical pores and the strengths of the surface field [7]. The volume ratio of the
blocks, the immiscibility of different blocks and interaction between the confining
surfaces and the blocks are important factors on which the morphology of diblock
copolymer confinement in a cylindrical tube rests [9]. Block copolymers confined by
cylindrical pores using Cartesian coordinates have been studied with respect to the
dimensions of confinement, the diameter of the pore and domain spacings of micro-
domains. There was subtle competition observed between topological constraints and
chain stretching in the cylindrical phase [22]. The constrained frustration instigated by
confinement, leads the system into the intricate free energy state and in addition,
8
confinement frustration tends to orientate microstructures perpendicular to the pore
surface. Furthermore, the system of weak segregation can easily be deformable under
the cylindrical confinement, the phase behaviour of such a system is more complex as
compared to a rigid system and such a system may be used to produce more delicate
materials [10]. The Lamella forming system confined in cylindrical tubes, the
competition between the incommensurability of the diameter of the tube and the
lamellae period and preference of the surface for the A segments are important for the
comparable values of diameter and period [13]. Theoretical studies show that
nanostructures for the development of the nano-devices can be accomplished by the
self-assembly of diblock copolymers in a confined geometry [23]. Diblock copolymer
melts confined to relatively narrow cylindrical pores of different diameters explored a
sequence of novel structures as compared to bulk, which form with variation in pore
diameter. These sequences of patterns were insensitive to the pore wall interaction [24].
Figure 2.1: The bulk phase diagram of diblock copolymer system [25]. The horizontal axis shows volume
fraction of A segment of polymer system, vertical axis on the left illustrates Flory-huggins interaction
parameter and vertical axis on the right presents the decreasing temperature parameter. In the diagram
(LAM) indicates lamella phase, (HEX) indicates hexagonal packed cylindrical phase, (BCC) is for body-
centred-cubic spheres phase and (GYR) refers to the bi-continues complex gyroid phase in the bulk
system.
The bulk system of diblock copolymer is a system which is large enough so that no size
effect can take place in the system. Hence, based on this definition the phase behaviour
9
of the bulk system of AB diblock copolymer depends on three experimentally
controllable factors volume fraction of A segment f, degree of polymerization N and
Flory-Huggins interaction parameter . Note that the product N is inversely
proportional to the temperature parameter [26].
In the bulk system under equilibrium conditions, the minor segment segregates from the
major segment of a diblock copolymer to form regular and uniformed spaced
nanodomains [27]. The segregated shapes are mostly forms due the volume fraction of
the minority segment of the diblock copolymer system Af and by the chemical
incompatibility of the both segments. The body-centred cubic sphere phases and
hexagonally packed cylinders phases forms when the volume fraction of minority
segment is in the range of 40.06.0 Af in the system. The alternating lamella phases
form based on the choice of the volume fraction for the minority block in the range of
60.045.0 Af within the nanostructures. However, various computational [28-30]
and experimental [31-33] studies fixed the value of volume fraction by tuning it at
40.0Af for spheres and cylindrical phases. While, to obtain the lamella phases these
two values 48.0Af for asymmetric lamella and 50.0Af for symmetric lamella are
used in the melt. At low temperature spontaneous ordered nanostructures forms in the
melt. The phase diagram of diblock copolymers shown in Figure 2.1 suggests that the
sphere phases form on very low temperature for the values less or equal to 0.25 (
25.0 ), the cylindrical phases form for the values of temperature parameter 30.0
and for the alternating lamella phases the values are less than 0.35 ( 35.0 ). The rest
of CDS parameters are derived and fixed by tuning in the mathematical and
computational studies [30, 34-36]. Note that rest of the CDS parameters are not derived
from bulk phase diagram and are discussed in chapter three.
10
Figure 2.2: AB diBlock copolymers nanostructures in the bulk systems [22].
The study of soft matter is associated with the macroscopic mechanical properties of
materials such as colloids, surfactants, liquid crystals, biomaterials, polymers in the melt
or solution and in addition, several soft materials can be induced to flow under certain
parameters [37]. Many polymer blends are highly immiscible and segregate into
different phases. In many applications it is required to combine the properties of
different polymers in a blend. In the polymer blends, phase separation occurs in
microscopic length scale as well as macroscopic length scale. Meanwhile, AB diblock
copolymers contain two blocks of different polymers interconnected linearly to each
other and are less miscible on cooling; in addition phase separation occurs due to their
chemical incomparability [38, 39]. The length-scale for phase separation in block
copolymers is prevented by the connectivity of blocks which is about 10-100nm as a
contrast with blends of polymers. The tendency of blocks to segregate on lowering
temperature is called micro-phase separation. Block copolymers, due to microphase
separation form variety of ordered structures in the bulk systems [37] such as lamella,
hexagonally packed cylinders, body-centred cubic arrays of spheres, double gyroid, and
double diamond as shown in Figure 2.2. Many smaller structural units of monomer
linked covalently in any conceivable pattern form a large polymer molecule. A small
molecule is said to be a monomer or “building block” if it has bonding ability (has a
bonding site) with another monomer to form the polymer chain. The numbers of
bonding sites are called functionality. The monomers with bi-functionality form linear
molecules whereas the monomers with poly-functionality have three or more bonding
11
sites. The homopolymer is made up of a macromolecule which is developed by a single
species of monomer, whereas, the copolymer is composed of two types of species of
monomers. The scope of the topic is only limited to the polymers made up of bi-
functional monomers. The architecture of copolymers can be divided into five grafting
types which are A-B diblock copolymer, A-B-A triblock copolymer, Grafted A-B
diblock copolymer, Star A-B-C triblock copolymer and linear A-B-C triblock
copolymer, schematic diagrams of various grafting types block copolymers are shown
in Figure 2.3.
Figure 2.3: Schematic diagrams of different architectures of block copolymers. The green represents the
block A, red one represents the block B and the blue one represents block C. (a) A-B diblock copolymer,
(b) A-B-A triblock copolymer, (c) Grafted or branched A-B diblock copolymer, (d) Star A-B-C triblock
copolymer, (e) Linear A-B-C triblock copolymer.
The self-organization property in the soft materials is a primary mechanism which leads
the nanostructure formation. Soft materials with the ability of self-organization have
received considerable attention in the nanotechnology over the last two decades as an
efficient means of patterning surfaces on mesoscale for real applications [40, 41]. The
applications include construction of high-capacity data storage devices, waveguides,
quantum dot arrays, dielectric mirrors, fabrications of nanoporous membranes for
advanced separation media and nanowires [12, 42]. Block copolymers are
12
macromolecules composed of chemically different blocks and are covalently connected
to each other. They are an important class of materials with the natural ability to self-
assemble into various periodic structures in the range of 10-100nm [43, 44]. At high
temperatures, entropy dominates and consequently both blocks mix together
homogeneously i-e form a disordered state and by lowering temperature and
correspondingly increasing the Flory-Huggins interaction parameter, the micro-phase
separation of A and B blocks may adopt different morphologies. The parameters that
determine the micro-phase separation in the bulk system are the Flory-Huggins
interaction parameter , the degree of polarization N and the volume fraction of A
monomer Af [45]. Depending on the composition of the copolymer and in the variation
of N , the micro-phase separated structure forms various patterns in the bulk systems.
The Bulk phase diagram of diblock copolymers has been studied and understood on the
basis of experiments and theoretical studies [46]. A number of classically ordered
equilibrium phases have been identified in diblock copolymer melts such as lamellae,
hexagonally packed cylinders, a body-centred array of spheres, and a bicontinuous
network structure gyroid [47]. The control of long range order in structure is very
important for practical applications such as the use of block copolymers in electronic
and photonic applications require the production of highly ordered and defect free-
structures.
2.2. Block copolymers in planar thin films
Due to its fascinating property of self-organizing, diblock copolymers are very
important for fabricating materials for smart nano bioreactors and self-regulating
diagnostic tools. To accomplish these goals nanostructure control is needed. Cylinder
forming block copolymers are being used as templates for fabricating parallel stripes in
nanotechnology [48], such as templates for nanowires [49] and polarizers [50].
13
Technological applications require the use of block copolymers in thin film geometries,
where self-assembly is strongly influenced by surface energetics [51]. To explore new
morphologies as compared to the bulk system, confinement is influential apparatus in
breaking the symmetry of the nanostructure [28, 52]. The Cell Dynamic System method
was employed to study diblock copolymers in planar thin films under the influence of
moving walls on lamellae forming system. This study revealed that well-aligned
lamellae form parallel to the walls for larger values of the surface interaction and
perpendicular to the walls for a small value of the surface interaction obtained [53]. The
mean-field model has also been used to study the dynamics of microphase separation
and orientation of lamellae in two dimensions. In this investigation formation of
lamellar structure and their stable states are discovered which depends on both periods
of the lamellae and minimum of the free energy. The influence of the confining plane
boundaries and interaction with block copolymers on the lamellae orientation was
investigated using free energy calculations. Variation with respect to boundaries of the
surface property at one boundary, the transition from preferential perpendicular to the
parallel lamellar orientation was obtained [54]. Directional quenching also can influence
the growth of oriented microphase-separated structures as confinement and shear flow,
this observation has been verified using the CDS method applied in two dimensions in
which the moving boundary resulted in ordered and disordered regions. With
sufficiently slow boundary speed, well-aligned lamellae but with defects were obtained,
On the other hand, with a too high boundary speed an irregular morphology which
resembles with the homogeneous quench was obtained. Regular patterns of hexagonal
structure are reported to be generated by low boundary speeds whereas with high speeds
perpendicular lamellae to the growth front are generated subject to the small thermal
fluctuation in the disordered region in the front of the boundary. Increasing both
amplitudes of thermal fluctuations and boundary speeds, the formation of the hexagonal
14
structure was obtained [55]. The characteristic finger print morphology was obtained in
planar thin films. In this study, alignment of the perpendicular lamella and reorienting
the lamellae in the direction of shear was obtained by annealing the films above the
glass transition temperature of the two blocks in the presence of an applied shear stress
[56].
2.3. Block copolymers confined in cylindrical pores
The effect of confinement in pattern formations of copolymer systems is an area of
significant importance in the polymer research. Sevink et al investigated the influence
of pores on the pattern formation of symmetric diblock copolymers confined in the
cylindrical pores, they showed under various pore sizes two types of patterns depending
on the surface interactions: one is perpendicular and other is parallel patterns. Through
variation of the pore radii and volume fractions, no mixed parallel/perpendicular
patterns were observed in their investigations and the obtained patterns were unaffected
by the pore diameter. The perpendicular pattern has slow kinetics and has a helical
structure as an intermediate depending on their pore size. The kinetics of parallel pattern
was observed unaffected by the pore radius for various ranges under considerations
[17]. The systematic study has been carried out by confined assembly of silca-surfactant
composite nanostructure in cylindrical pores with variation in pore diameter. With the
same reaction conditions and precursors that were used for the production of the two-
dimensional hexagonal SBA-15 nanostructured thin films, exceptionally silica
nanostructured patterns with single and double helical geometries instinctively observed
inside individual alumina cylindrical nanopores. In addition, by narrowing down the
degree of the confinement, the transition from coiled cylindrical to the spherical cage-
like geometric patterns was observed in cylindrical nanopores [16]. The lamella forming
system was investigated under geometric confinements [57], results show concentric
15
lamellae sheets, bent lamellae pointing towards the column mantle and parallel to the
pore sheets. The various patterns of symmetric A-B diblock copolymer system under
geometric confinement in cylindrical channels were investigated by using Monte Carlo
simulation method. The studies were carried out by considering the influence of
incommensurability of the pore diameter, lamella period and volume fraction of one of
the monomers of the polymer system. The simulation results reported in this study
includes the formation of lamellae normal to the cylinder axis, circular lamellae, porous
lamellar (mesh) pattern, lamellae parallel to the cylinder axis, single and double helixes.
However, using the Monte Carlo simulation method the exact value of lamellae period
cannot be obtained and commensurability of the simulation box and lamellae period was
crucial to determine the exact value for lamellae period so therefore approximate value
for lamellae period was used in simulations [13]. A systematical study of self-assembly
of diblock copolymers confined in cylindrical channels was investigated using the
simulated annealing method. Simulation results for diblock copolymers reported in
cylindrical pores are 2D hexagonally packed cylinders, helices, stacked toroids. The
obtained patterns of diblock copolymers were dependent on the pore diameter and the
surface-polymer interaction, structural frustration and interfacial interactions. The
degree of structural frustration parameterized by the ratio of pore diameter and lamellae
period length is crucial for understanding the mechanism of pattern transitions and in
addition they observed the transitions from helices to toroids to the sphere by imposing
a tight degree of confinement [14]. The confined diblock copolymer was investigated by
dissipative particle dynamics (DPD) and the results shown indicate patterns dependent
on the volume fraction one of the block, the pore diameter of the cylinder, the
interaction between the blocks and interaction between the wall and blocks. This study
was carried out for both symmetric and none symmetric lamellae. According to the
reported results perpendicular lamellae or stacked disc pattern formed in general for the
16
symmetric diblock copolymers under the condition that the cylindrical wall was kept
uniform towards the both blocks. However, in the exceptional case of small pore
diameter, a special bi-helix pattern was discovered due to the influence of entropy. In
the case when the cylindrical walls kept non-uniform and pore diameter was increased
transition from perpendicular-parallel-perpendicular lamellae observed, in addition, the
increase in the non-uniformity of the cylindrical wall didn’t respond in the shape other
pattern but just only parallel lamellae formed. In the case of asymmetric lamellae multi-
cylindrical micro-domains, multilayer helical phases and other complex patterns were
observed [9]. The confinement of the self-assembled diblock copolymers in the
cylindrical nanopores was investigated using lattice Monte Carlo simulations and the
simulation results obtained by variation of pore diameter and strength of the surface
field. The simulation results for diblock copolymers shown are stacked-disk, single
helix, gyroidal, catenoid-cylinder, stacked-circle and concentric cylindrical barrel
patterns [7]. Later Wang et al using the same method lattice Monte Carlo simulation
method applied on symmetric diblock copolymer by systematic variation in pore
diameter and the surface preference of the A monomer. The pore center can be either A
or B monomer which depends on the size of pore diameter while the A segment
segregates due to the strong surface preference and at neutral or weak surface
preference lamellae perpendicular to the pore axis obtained regardless the size of the
pore diameter, in addition, small surface preference patterns were undulated and
furthermore by increasing surface preference the concentric cylinders obtained.
However again it was crucial for a good estimate of the bulk period and efficient
sampling of system configuration [6]. The self-assembly of asymmetric diblock
copolymers were also investigated using self-consistent field theory (SCFT) under
cylindrical confinement, the parameters 14NAB and 36.0Af were used in the
simulations [10]. A variety of nanostructures have been obtained using confinement
17
dimension. The system could freely achieve local free energy minimum when the pore
diameter is commensurate with polymer period under surface preference and
accommodation with confined pore diameter and on the other hand, the system can
achieve a complex free energy landscape state due to the frustration caused by
confinement in the incommensurate case. Furthermore, by tightening the degree of
confinement frustration leads to orientate structures perpendicular to the pore surface
and therefore study suggest that system is easily deformable at relative weak
segregation under cylindrical confinement [10]. Diblock copolymer studies were
investigated by using lattice simulated annealing method in cylindrical nanopores under
confinement [21]. Also in this study the pore diameter and surface preference were
systematically varied to investigate effects on structure formation and chain
conformations. The study reveals the novel structures for lamellae forming system and
cylindrical forming systems such as helices and perforated lamellae. Studies also show
that the chains near the pore surfaces are compressed relative to the bulk chains which
show the existence of the surfaces [21]. The real space self-consistent field theory
method applied for the systemic study of diblock copolymers confined under relatively
narrow cylindrical nanopores with various pores sizes. This computational study shows
various structures in cylindrical forming systems obtained by increasing pore diameter
including a single cylinder, stacked discs, single helix, double helix, toroid-sphere and
helix-cylinder. All of these obtained patterns were sensitive to the pore wall interaction
[24]. The dynamic density functional theory was employed to study confined diblock
copolymers melt in cylindrical nanopores. In this study confined was two-dimensional
and obtained results are general which includes catenoid, helix, the double helix and
structure which resembles with experimental toroidal structure. This computational
study also suggests that the patterns are sensitive to weak surface fields and small pore
radii variations, in addition, these pore radii variations become responsible for interfaces
18
between the stable patterns and these interfaces become responsible for the transition for
other patterns as well [58]. The computational study and analysis through CDS method
were carried out systematically on diblock copolymer lamellae, cylindrical and
spherical patterns. In this study various patterns of diblock copolymers were reported
and all the structures were obtained by variation of four factors: pore diameter, affinity
of pore wall to the one of the block, incommensurability of the pore lengths and domain
spacing [59]. The simulated annealing technique was employed to investigate the
influence of the confining geometries on self-assemblies of diblock copolymer
cylindrical forming system in cylindrical, spherical and ellipsoidal nano-pores [43].
2.4. Block copolymers confined in spherical pores
Block copolymers are a splendid class of materials which are gaining scientific interest
owing to their self-assembling property and potential applications [60-62]. Block
copolymers can be helpful in designing smart bioreactors on mesoscale and drug
delivery nano particles [63, 64]. Recently, block copolymer confinements in spherical
pores have been receiving extensive consideration among the scientific community. To
obtain the nano structural control, confinement is considered to be a powerful tool
which is very helpful in breaking the symmetry of the nano structures and favouring
materials to self-assemble into newly-ordered structures compared to the bulk [8, 31].
Furthermore, spherical confinement is a more effective confining geometry which is to
be considered. Block copolymers have been studied and investigated depending on the
surface field interactions and film thickness [65-67]. They reported perforated lamella,
parallel, perpendicular to the surface and cylinder. In the case of block copolymers
confined by the surface of the sphere, in addition to the surface field interaction, the
curvature influences the morphologies. The onion-like nanostructures and hexagonally
packed cylinders are prepared by variations in the block ratio and the concentration of
19
the solution for lamella forming systems [68]. Rider et al investigated lamella and
cylindrical forming morphologies under 3D confinements by experimental work; they
reported a variety of the nano structures such as onion-like lamella, complex golf ball
morphologies and spheres on the curved surfaces [31]. The study and investigation of
phase separation for diblock copolymer melt in a spherical geometry is a difficult
problem. To study phase-separated patterns for diblock copolymers on the surface of a
sphere, the study carried out using a finite volume method, study revealed stable and
integral defected structures which would not occur on a planar surface [69]. The defect
formation in ordered structures on curved surfaces is a hot topic, attracting growing
interest. The Self-consistent field theory simulation method was applied to thin block
copolymer melt films confined in spherical nano-pores. The study reveals that there is
delicate competition between topological constraints and chain stretching in cylindrical
phases, whereas in a lamellar phase with small sphere radii, a stable hedgehog defect
configuration was observed and for larger sphere radii, competition between hedgehog
and spiral defect conformation was noticed. A quasi-baseball formation with defective
spiral-like nano structure was found to be metastable [22]. The CDS method in
Cartesian coordinate system was used to investigate the morphology of thin block
copolymer films around a nano-particle. The reported structures in spherical nano-pores
for lamellae forming system are parallel, perpendicular, mixed and perforated lamellae,
whereas cylindrical forming systems were parallel and perpendicular cylinders. Studies
suggest that one could obtain nano-shells by using effective interplay of these
parameters which are the film thickness, the block copolymer characteristics, the
surface parameter and the curvature [20]. The CDS method was carried out by a
systematic study on thin block copolymer films around nano-particles to investigate
lamellae, cylinder and sphere morphologies. Investigations show various structures such
as standing lamellae, cylinders, onions, cylinder knitting balls, golf ball, layered
20
spherical virus-like and mixed morphologies with T-junctions and U-type defects [8].
There was also a computational study on two-dimensional block copolymers on the
surface of a sphere [70]. These computational studies were conducted by the square
lattice [8, 70] and triangular lattice [69]. However, in this contribution studies were
carried out on the confinement and the curvature-induced morphologies of block
copolymers using spherical lattice.
21
CHAPTER THREE
3. Implementation of CDS method in polar, cylindrical and
spherical coordinate systems
3.1. Overview
Prediction of the structure formation in copolymer melts by experiments is very difficult
and time-consuming but mathematical modelling can help to understand structure
formation. Molecular dynamics or Monte Carlo simulations can be used to study their
behavior but due to computational demands studies have been restricted to short chains
or low grafting densities. There are studies based on Self-Consistent Field Theory SCFT
in one dimension for morphologies with no angular dependence and in two dimensions
for morphologies with either rotational symmetry or without any radial dependence but
to perform full three-dimensional calculations is still a challenge for researchers [42].
To reduce computational costs the Unit Cell Approximation (UCA) have been routinely
used following SCFT to study bulk systems but accuracy was lost and the effects of
packing frustration over sighted were reported, which are both important for
nanoparticles [71]. However, the CDS is relatively simple Landau-Ginzburg type
simulation allows for dynamic and fast simulations of very large block copolymer
systems. Stiffness, architecture, polydispersity and density effects of the polymer chain
can only be indirectly modeled and the modification of the coefficients can be carried
out in the Landau free energy model. The CDS was originally developed to model the
interface dynamics in the phase separating systems. The CDS is a coarse grained
modelling in which it is possible to observe the microphase separation phenomenon in
the block copolymer systems and obtained nanostructures by CDS are comparable with
22
experimental results [72]. The CDS is efficient and accurate in investigating often very
complex dynamical behavior in a large set of systems. CDS predict the block copolymer
morphologies in generic agreement with other simulation methods and experiments, in
addition, some cases are in quantitative agreement. Being fast and large-scale CDS can
successfully describe time evaluation of morphologies in several cases, and is capable
of predicting new kinetics and the work-derivative properties of materials [12]. Pinna et
al [59] studied diblock copolymers in cylindrical pores using CDS in the Cartesian
coordinate system and they showed that a simple Ginzburg-Landau type theory can
predict a tremendous rich “zoo” of diblock copolymer morphologies in cylindrical
nanopores. Furthermore, CDS replicates details of a very complex collection of
confinement induced nanostructures which were previously done by slow simulation
techniques. In the presence of curvature and confinement, Ginzburg-Landau theory
works successfully. Although many computational techniques have been developed and
carried out, the field offers a large number of open questions, for some of which are
addressed in the thesis. To study and analyze accurately block copolymers on curved
surfaces, the computational model needs to be implemented in the curvilinear
coordinate systems.
Continuous physical space discretization into a uniform orthogonal computational space
is basic requirement for finite difference schemes and application of boundary
conditions needs to fall on coordinate surfaces of the coordinates system. Furthermore
for accuracy grid points need to be clustered in space of large gradients, whereas for
computational economy grid points must be spread out in space of small gradients. For
these requirements Cartesian coordinate system is incompatible [73], however, polar,
cylindrical, spherical coordinates are viable. In the finite difference schemes for the
partial differential equations some procedure is followed to discretise the space
coordinates with time axis, regardless of the dimensionality. However, for various
23
realistic physical problems, the basic choice Cartesian layout with square or rectangular
domains is not the natural environment. The suitable coordinate transformation of a
physical problem with partial differential equations is very difficult to find in most cases
[74]. The best option in this case is that to solve the partial differential equations on the
geometrical irregular domains. The order parameter which represents time evaluation of
block copolymers is used in CDS which varies continuously with respect to coordinate
r. There is a Partial Differential Equation (PDE) which represents the time evaluation of
order parameter with respect to each lattice point in the CDS method. The
corresponding Partial Differential Equations of CDS model was solved on the lattice by
taking account of each cell of the curved lattice. The discretization of Laplacian
operators were carried out in polar, cylindrical and spherical coordinate systems along
with appropriate boundary conditions. The lattice and weights commonly used in lattice
hydrodynamics simulations are computationally efficient discrete representations of the
Laplacian by means of conserving isotropy up to the leading order error. For the CDS
method these Laplacians should prove beneficial, as for other problems where efficient
isotropic discretization of Laplacian is required [75]. In the standard sense CDS model
is not a PDE solved by numerical analysis. However its accuracy depends on time steps
and spatial steps, therefore large time steps and large spatial steps cannot give accurate
solutions for the initial value problem of Cahn-Hilliard-Cook (CHC) equation. The
macroscopic observables are still quantitatively accurate which implies the concept of
qualitative accuracy of numerical analysis. In this sense, we can say that the scheme is
qualitatively accurate if the scheme can give quantitatively accurate results for the
quantities which are qualitatively characteric of the system [76]. On the basis of this
concept, we can say that the discretization scheme for the CDS model is a qualitatively
accurate discretization method for the CHC equation. In this contribution, a
computational study was carried out for diblock copolymers confined in circular annular
24
pores, cylindrical pores and spherical pores using curvilinear coordinate systems. The
CDS method is implemented by using curvilinear coordinates systems. The CDS is a
very good resolution between computational speed and physical accuracy.
3.2. CDS and its implementation in polar, cylindrical and spherical
coordinate systems
In the CDS method, time dependent variations of the order parameter with
respect to the coordinate system i and it can represents the concentration of the one
monomer in a binary blend. Computations of the order parameter with respect to time
were carried out by the partial differential equation called Cahn-Hilliard-Cook (CHC)
and this partial differential equation is given in terms of the free energy functional
which controls the tendency of local diffusion motion. The discretization of Laplacian
operator in curvilinear coordinates is carried out in this study. The Order Parameter
),( it is determined at time t and in cell i of a discrete lattice.
In CDS method of diblock copolymers, the compositional order parameter in terms of
local & global volume fractions is defined by the following relation [12]:
)21()()()( frri BA , (3.1)
where BA , are the local volume fraction of the monomers A and B respectively. The
volume fraction of monomer A is defined by the relation )/( BAAA NNNf , similarly,
the volume fraction of monomer B is defined by the relation )/( BABB NNNf , where
AN represents the number of monomers of the block A and BN represents the number of
monomers of the block B. BA NNN is the total degree of polymerisation. The
constant f represents here the block length ratio in the polymer system, note that if
taking symmetric block ratio in this case f=1/2 then the order parameter will take the
simple form BABA ))2/1(21( .
),( it
25
The change with respect to time t in the order parameter depends on the chemical
potential ),( tr and the mass current or a flux ),( trj which is linearly related to the
local chemical potential and which determines the dynamics of the order parameter
through the continuity equation given below [72]:
),(),(
trjt
tr
(3.2)
The gradient of chemical potential related to the mass current given below:
)},({),( trMtrj , (3.3)
where M is positive Onsager coefficient which describes the mobility of the monomer A
with respect to the monomer B. The chemical potential determined by the functional
derivative of free energy functional }{F with respect to order parameter is given in the
following mathematical relation:
}{),(
Ftr (3.4)
From equations (3.2), (3.3) and (3.4) one can easily derive the mathematical relation for
the order parameter which represents the change in the order parameter with respect to
time, and is called Cahn-Hilliard-cook (CHC) equation given below [77]:
][2 FM
t, (3.5)
where M the is phenomenological mobility constant and can be taken M=1 for
simplicity and for correspondingly setting the time scale for diffusive process. In
equation (3.6) ][F is the free energy functional, which acts as a global Lyapunov
functional and tends to decrease with respect to time towards its minimum and which
can be written as below by dividing it by kT [72]:
)()()(2
]2
)([)]([2
rrrrGrddrBD
HdrrF (3.6)
26
In equation (3.6) on the right hand side, the first term represents the short range
interactions whereas, the second term with double integrals represents the long-range
interactions due to the connectivity of the sub-chains. In the above equation (3.6) the
term 2/|| 2D , generates free energy necessary to create an interface between the
segments A and B of the polymer system. The Green’s function ),( rrG satisfies
Laplace’s equation )(),(2 rrrrG . In the equation (3.6) D represents the
diffusion constant, and B represents the chain length dependence to the free energy. The
term )(H in the above equation (3.6) accounts the local contributions to the free energy
and can be expressed in various expansions; however, in accordance with Ginzburg-
Landau theory [72], following expansion was used in the current analysis:
4322
4)21(
3])21(
22[)(
uff
AH , (3.7)
where represents temperature and uandA ,, are phenomenological constants, these
all are parameters which can be related to molecular characteristics [12]. Ohta and
Kawasaki explained that ,)21( 2fA D and B are related with the degree of
polarization N, chain length b and the Flory-Huggins parameter . Here b is Kuhn
segment length of the polymer. The mathematical relations are given by [78]:
)1(48
,)1(4
)(
2
1 2
22 ff
bD
ff
fsN
N
, and
2222 )1(4
9
ffbNB
. (3.8)
The Flory-Huggins parameter is inversely proportional to the temperature, if it is
positive then it measures the strength of repulsion between incompatible blocks, if it has
negative value it reflects the free energy which stimulates blocks towards mixing. In the
set of equations (3.8), s(f) is an empirical fitting function of order 1. For simplicity we
adopted the notations of D and B by replacing D~
, B~
respectively. The parameters u and
27
v can be calculated by the vertex function which is given by Leibler [79]. For simplicity
these very complex functions are replace by approximate constants because the
phenomena under consideration is quite general so that it is permissible for choosing the
parameters for equations (3.6)-(3.7) as phenomenological constants [28].
Numerical evolution of Equation (1) by means of CDS is given below.
),(),(),()1,( titittiti , (3.9)
where ),(),( titi , is the isotropic discrete Laplacian operators in polar,
cylindrical or spherical coordinate system for the quantity ),( ti . In the equation (3.9) i
represents iii r , for polar coordinate system, ),,( zr iiii for cylindrical coordinate
system and )( iiii r for spherical coordinate system. In the equation (3.9) t shows the
time steps taken in the current analysis. The following relation was used in the study for
the calculations of free energy functional:
),()],(),([),()),((),( tiBtitiDtitigti (3.10)
For the case when polymers were confined between two interfaces then the above
equation for free energy functional was modified as following:
)()],(),([),()),((),( rstitiDtitigti i , (3.11)
whererrr Nnorniii hrs 1)( , i denoting the attracting block (either A or B) of block
copolymer system, ih is the strength of interaction between confining walls and the
blocks of polymer system and ba is Kronecker delta [80].
Where so-called map function is given below:
322 )21(])21(1[)( uffAg (3.12)
In the next section, the Laplacian operators are discretised in the polar, cylindrical and
spherical coordinate systems. The Laplacian operators were used in the computations of
equations (3.9) and (3.10) where averaging of the stencil of lattice was required.
28
3.3. Discretisation of Laplacian in the polar coordinate system
Figure 3.1: Polar mesh diagram.
The conservative form of Laplacian operator in polar coordinates is given below:
2
2 11
rrrrr (3.13)
20& ba rrr
For the discretisation of the Laplacian operator in polar coordinate system, Forward
Time Central Space (FTCS) explicit finite differencing scheme was used in the current
analysis.
1,,1,22
,1,1,1,,12
2
2)(
1
)()(2
1)2(
)(
1
jijiji
i
jiji
i
jijiji
r
rrr
(3.14)
Here domain is circular mesh contained between two circles of radii ar and br , where
rirr ai and jj for NjNi r ,...,2,1,,...,2,1
Isolating the term ji, get the following form:
ji
i
ijiji
i
jiji
i
jiji
rr
rr
r
rrr
,222
222
1,1,22
,1,1,1,12
2
)()(
)()(2)(
)(
1
)()(2
1)(
)(
1
(3.15)
Now
)()( 2211
2 nn hOhOS (3.16)
29
Where S denotes a stencil in polar coordinate system and terms )()( 2211
nn hOhO are the
truncation errors of the order n due to finite mesh sizes 21,hh . The stencil must satisfy
following condition [81].
3
1
3
1
, 0k l
lkS (3.17)
It follows that:
0)()(
)()(2)(
)(
1
)()(2
1)(
)(
1
,222
222
1,1,22
,1,1,1,12
ji
i
ijiji
i
jiji
i
jiji
rr
rr
r
rrr
(3.18)
Multiplying both sides by of equation (3.18) by the term ))()((2
)()(222
222
rr
rr
i
i
we have
the following equation:
0
)()(
1
)()(2
1)(
)(
1
))()((2
)()(,
1,1,22
,1,1,1,12
222
222
ji
jiji
i
jiji
i
jiji
i
i
r
rrr
rr
rr
So therefore, finally isotropised discrete five point Laplacian in polar coordinates can be
written as in the following form:
rr N
i
N
j
ji
N
i
N
j
jiji
i
jiji
i
jiji
r
rrr
1 1
,
1 1
1,1,22
,1,1,1,12
)()(
1
)()(2
1)(
)(
1
(3.19)
Where ))()((2
)()(222
222
rr
rr
i
i
is weighting factor for the nearest neighbourhood points
in the polar mesh system.
The following periodic boundary conditions were applied on the angular coordinate of
the polar mesh for evaluation of the Laplacian operator in polar coordinates:
)1,()1,( iNi
, ),()0,( Nii , for all rNi ,.....,2,1 .
30
The following reflective boundary conditions were applied on the radial coordinate for
calculations of the Laplacian in polar coordinate system:
,),(),1( jNjN rr ),1(),0( jj , for all Nj ,....,2,1 ,
where NjNiji r ,...2,1,,...,2,1),,( are approximations of the order parameter
function ),( r . The initial condition for the system is chosen to be the random
distribution of the of the order parameter )5.0,5.0(),( 0 tr .
3.4. Discretisation of Laplacian in the cylindrical coordinate system
Figure 3.2: Cylindrical mesh diagram [82].
The Laplacian in cylindrical coordinates is given below.
2
2
2
2
22
22 11
zrrrr
, (3.20)
for boundary conditions hzrrr ba 0&20, . Finite difference scheme
for the above Laplacian in cylindrical coordinates can be written as below:
]2[)(
12
)(
1
)()(2
1)2(
)(
1
1,,,,1,,2,1,,,,1,22
,,1,,1,,1,,,,12
2
kjikjikjikjikjikji
i
kjikji
i
kjikjikji
zr
rrr
(3.21)
31
Here domain is cylindrical mesh contained between two circles of radius ar and br where
rirr ai , zkzj kj , for zr NkNjNi ,...2,1,,...,2,1,,...,2,1
Isolating the term ji, get the following form:
kji
i
kjikjikjikjikjikji
i
kjikji
i
kjikjikji
zrr
zr
rrr
,,2222
1,,,,1,,2,1,,,,1,22
,,1,,1,,1,,,,12
2
)(
2
)(
2
)(
2
]2[)(
12
)(
1
)()(2
1)2(
)(
1
or can be re-written as below:
kji
i
ii
kjikjikjikjikjikji
i
kjikji
i
kjikjikji
zrr
rrzrzr
zr
rrr
,,2222
22222222
1,,,,1,,2,1,,,,1,22
,,1,,1,,1,,,,12
2
)()()(
)()()()()()(2
]2[)(
12
)(
1
)()(2
1)2(
)(
1
Now
)()()( 332211
2 nnn hOhOhOS (3.22)
Where S denotes a stencil in cylindrical coordinate system and terms
)()()( 332211
nnn hOhOhO are the truncation errors of the order n due to finite mesh sizes
321 ,, hhh . The stencil must satisfy following condition [81]:
3
1
3
1
3
1
,, 0k l m
mlkS (3.23)
It follows that:
32
0)()()(
)()()()()()(2
]2[)(
12
)(
1
)()(2
1)2(
)(
1
,,2222
22222222
1,,,,1,,2,1,,,,1,22
,,1,,1,,1,,,,12
kji
i
ii
kjikjikjikjikjikji
i
kjikji
i
kjikjikji
zrr
rrzrzr
zr
rrr
(3.24)
Multiplying Equation (3.24) both side by
22222222
2222
)()()()()()(2
)()()(
rrzrzr
zrr
ii
i and let say:
0
)2()(
1)2(
)(
1
)()(2
1)2(
)(
1
,,
0
1,,,,1,,2,1,,,,1,22
,,1,,1,,1,,,,12
1
kji
kjikjikjikjikjikji
i
kjikji
i
kjikjikji
zr
rrr
Then equation (3.24) becomes:
0)()()()()()(2
)()()( 01
22222222
2222
rrzrzr
zrr
ii
i (3.25)
Now we can write isotropic discrete Laplacian in cylindrical coordinate system as
below:
01 NN
(3.26)
Where 22222222
2222
)()()()()()(2
)()()(
rrzrzr
zrr
ii
i is weighting factor for
isotropic form of the Laplacian in cylindrical coordinate system.
The following periodic boundary conditions were applied on angular coordinate in the
current analysis.
,),0,(),1,( kikNi
,),,(),0,( kNiki for all hr NkNi ,...,2,1,,...,2,1 .
33
The following reflective boundary conditions were applied on radial coordinate in the
cylindrical system.
,),,(),,1( kjNkjN rr ,),,1(),0( kjkj for all hNkNj ,...,2,1,,...,2,1
The following periodic boundary conditions were imposed on the coordinate z in the
cylindrical coordinate system.
),,()0,,( hNjiji for all NjNi r ,...,2,1,,...,2,1 .
Where hr NkNjNikji ,...,2,1,,...2,1,,...,2,1),,,( , are approximations of the
function ),,( hr . The initial conditions for order parameter were chosen to be the
random distribution of the )1,1(),,( 0 tzr .
3.5. Discretisation of Laplacian in the spherical coordinate system
Figure 3.3: Sphere grid system diagram [83]
The Laplacian in spherical coordinates is given below.
222
2
22
2
22
22
)(sin
)(cot12
rrrrrr (3.27)
Subjected to boundary conditions 20,0, ba rrr , finite difference
scheme for the above Laplacian is given below:
,)1,,()1.,( jiNji h
34
]2[))(sin(
1
)()(2
cot2
)(
1
)()(2
2)2(
)(
1
1,,,,1,,222
,1,,1,2,1,,,,1,22
,,1,,1,,1,,,,12
2
kjikjikji
kjikjikjikjikji
i
kjikji
i
kjikjikji
r
rr
rrr
(3.28)
Discretising of the spherical domain was carried by using the following conditions.
NkNjNiforkjrrrrirr rkjbaai ,...2,1,,...,2,1,,...,2,1,,,,
To obtain average value isolating the term kji ,, from above equation we have.
)())(sin(
1
)()(2
cot)(
)(
1
)()(
1)(
)(
1
1,,1,,222
,1,,1,2,1,,1,22
,,1,,1,,1,,12
2
kjikji
kjikjikjikji
i
kjikji
i
kjikji
r
rr
rrr
kji
iirrr
,,222222 )(sin
1
)(
1
)(
12
We can write above equation as follow:
)())(sin(
1
)()(2
cot)(
)(
1
)()(
1)(
)(
1
1,,1,,222
,1,,1,2,1,,1,22
,,1,,1,,1,,12
2
kjikji
kjikjikjikji
i
kjikji
i
kjikji
r
rr
rrr
kji
i
i
rr
rrr,,22222
222222222
sin)()()(
)()(sin)()(sin)()(2
Now
)()()( 332211
2 nnn hOhOhOS (3.31)
Where S denotes a stencil in spherical coordinate system and terms
)()()( 332211
nnn hOhOhO are the truncation errors of the order n due to finite mesh sizes
321 ,, hhh . The stencil must satisfy following condition [81].
(3.29)
(3.30)
35
3
1
3
1
3
1
,, 0k l m
mlkS (3.32)
Therefore we can write above equation in the following form.
)())(sin(
1
)()(2
cot)(
)(
1
)()(
1)(
)(
1
1,,1,,222
,1,,1,2,1,,1,22
,,1,,1,,1,,12
kjikji
kjikjikjikji
i
kjikji
i
kjikji
r
rr
rrr
0sin)()()(
)()(sin)()(sin)()(2 ,,22222
222222222
kji
i
i
rr
rrr
Multiplying both sides of equation (3.31) by the following term
222222222
22222
)()(sin)()(sin)()(
sin)()()(
2
1
rrr
rr
i
i and let say
kji
kjikji
kjikjikjikji
i
kjikji
i
kjikji
s
r
rr
rrr
,,
0
1,,1,,222
,1,,1,2,1,,1,22
,,1,,1,,1,,12
)())(sin(
1
)()(2
cot)(
)(
1
)()(
1)(
)(
1
The above equation takes the following form:
0)()(sin)()(sin)()(
sin)()()(
2
1 0
222222222
22222
s
i
i
rrr
rr
Now we can write isotropic discrete Laplacian in spherical coordinates coordinates as
below:
0 NN
s (3.34)
Where is weighting factor for isotropic Laplacian in spherical coordinates which is
given below:
(3.33)
36
222222222
22222
)()(sin)()(sin)()(
sin)()()(
2
1
rrr
rr
i
i
The following periodic boundary conditions were applied on the angular coordinate of
the spherical coordinate system.
,),1,(),1,( kikNi
,),,(),0,( kNiki for all NkNi r ,...,2,1,,...,2,1
The following reflective boundary conditions were applied on the radial coordinate of
the spherical coordinate system.
,),,(),,1( kjNkjN rr ,),,1(),,0( kjkj for all NkNj ,...,2,1,,...,2,1
The following periodic boundary conditions were imposed on the azimuthal coordinate
of the spherical coordinate system.
,),,()0,,( Njiji for all NjNi r ,...,2,1,,...,2,1
Where NkNjNikji r ,...,2,1,,...2,1,,...,2,1),,,( ,is approximation of the
function ),,( r . The initial condition for the system is chosen to be the random
distribution of the )1,1(),,( 0 tr .
3.6. Simulations of binary fluid
To test the method and developed CDS code, simulations of a binary fluid were carried
out in the annular circular pore system. The simulation results for the binary mixture
were obtained by using CDS parameter systems for lamella forming system except (B =
0). In this case, instead of microphase separation, a macrophase separation occurred in
the pore system. This simulation setup is useful to investigate the behaviour of the
mixture in the pore system. The simulation results were obtained on one million time
steps to get the minimum energy level of the system.
,)1,,()1.,( jiNji
37
Figure 3.4: Phase separation for binary fluid were obtained on 1 million time steps, in a pore system with
the interior radius of the pore system was fixed at ar = 3.0 and the grid sizes are (a) 30x360 (b) 60x360
The simulation result shows that domain clearly divide into two rich subdomains. In the
pore system shown in Figure 3.4(a), domain divided into two rich subdomains, where
A-rich subdomain is blue having its minimum 9090243.0 and B-rich subdomain
having its minimum 077136.1 . The pore system shown in Figure 3.4(b) also
divided into two subdomains, where A-rich subdomain has its minimum 908395.0
and B-rich domain has its minimum 083791.1 .
38
CHAPTER FOUR
4. Block copolymer system confined in circular annular
pores
4.1. Introduction
Two-dimensional studies of diblock copolymers are not sufficient to provide full
information about the phase separation and structural control. However, it provides a
strong background for three-dimensional studies and which can give reflections about
the phase separation and structural control in three dimensions. Therefore for this reason
in the first phase of the investigation, two-dimensional computational studies have been
carried out in circular annular pores using the polar coordinate system. The diblock
copolymer system is explored in an annular circular pore with various pore sizes in the
annular circular pore. All the CDS simulations were carried out using periodic boundary
conditions in the angular direction 20 , where the angular step is taken
360/2 , and reflective boundary conditions were applied in radial r direction.
The pore is contained between two concentric circles where the interior radius of pore
system is fixed at ar = 3, 5, 7 in three type of pore systems and exterior radius of pore
system was varied for different values of br to expand the pore size of the system.
Therefore, circular annular pore size becomes d = ba rr where d represents pore size
of the system. The simulation results were obtained on 1 million time steps. All
simulations were carried out by an initial random disordered state of order parameter
within the range of ± 0.5. In the CDS system, we are able to visualise simulation results
by showing both blocks (A and B) of polymer system in the computational domain.
Simulation results were obtained for the diblock copolymer system in computation
39
domain patterned in two rich domains i-e red rich domains represents A block and blue
rich domain represents B block of the polymer system. Computational results for the
diblock copolymer system are also obtained with surface preference with one of the
blocks of the polymer system. In this study, the parameter represents the strength of
interaction between the wall and one of the blocks of the polymer system. Using the
CDS method employed in the polar coordinates we investigated classical morphologies
of lamellae, cylinder, and sphere in annular circular pores system. Lamellae forming
system, cylindrical forming system, and spherical forming system are also investigated
by one-dimensional confinement and two-dimensional confinements. Following table
shows the computational parameters used in all simulations in polar coordinates system.
Table 1: Computational parameters used in the simulation results
Radial step
r
Angular step
Time step
t
Total time steps
0.1 0.017453292 0.1 1 Million
4.2. Results and discussions
In this section, the results for the diblock copolymers system confined in the circular
annular pores are presented and discussed. The results for diblock copolymers shown in
this section were obtained in the pore geometry with various film thicknesses and the
pore radii of the pore system. The results are also presented which were obtained with
interfacial circular walls having the affinity with both the majority segment and the
minority segment of the polymer system.
4.2.1. Asymmetric lamellae forming system confined in circular annular pores
In CDS system, lamellae morphology can be obtained using different global volume
fractions of f = 0.50 in the symmetric case and f = 0.48 in the asymmetric case with
same temperature parameter 36.0 . The global volume fraction play an important role
40
in morphology formation, therefore simulation results were obtained for the lamellae
forming system against both values of global volume fraction i-e f = 0.50 (symmetric)
and f = 0.48 (asymmetric). However, the rest of the CDS parameters were also tested
and modified accordingly. In the following section asymmetric lamellae forming system
is investigated in annular circular pore system.
I. Asymmetric lamellae forming system confined in neutral circular annular
pores
The simulation results for diblock copolymer lamellae morphology for asymmetric
global volume fractions (f = 0.48), in the circular annulus pore systems with various
pore sizes and pore radii were obtained and discussed. The results were obtained on 1
million time steps. The CDS simulations were performed with CDS parameter for
asymmetric lamellae system given in table (2).
Table 2: CDS Parameters for asymmetric lamella forming system
f u v B D A
0.36 0.48 0.38 2.30 0.02 0.70 1.50
The pore domains were contained between two concentric circles for which the interior
radius ar of the pore systems was fixed, while the pore sizes were expanded by
increasing the exterior radius br of the pore system. The interior radius of the pore
system was fixed at ar = 3.0 in all the following simulation results, however the
exterior radius of pore systems were varied by expanding exterior radii of pore systems
for different sizes br = 4, 5, 6, 7, 8, 9, 11, 13 hence film thicknesses under investigation
were d=1, 2, 3, 4, 5, 6, 8, 10.
41
Figure 4.1: Evolutions of asymmetric lamellae system on 1 million time steps confined in various circular
annular pore sizes with the interior radius of the pore was fixed at 0.3ar and the exterior radius of pore
was expanded to obtain various pore sizes: (a) evolution of asymmetric lamella in the pore system size d
= 1, pore system induces alternating lamella strips (b) evolution of asymmetric lamella in the pore system
size d = 2, system patterned into Y-shaped, U-shaped and perforated holes morphologies (c) evolution of
asymmetric lamella in the pore system size d = 3, pore shows Y-shaped or tilted Y-shaped star shaped
and W-shaped patterns (d) evolution of asymmetric lamella in the pore system size d = 4, system induces
Y-shaped, W-shaped U-shaped and a few perforated holes (e) evolution of asymmetric lamella in the pore
system size d = 5, patterns are two arm stars and single arm star with a perforated hole at the centre of the
star (f) evolution of asymmetric lamella in the pore system size d = 6, induced nanostructures include
perforated holes morphology (g) evolution of asymmetric lamella in the pore system size d = 8, mixed
morphologies (h) evolution of asymmetric lamella in the pore system size d = 10, alternating lamella
strips normal to the exterior circular boundary.
The lamellae system in circular annular pore system shows a mixed orientation of strip
lamella, parallel to the pore system, perpendicular to the pore system and oblique.
Lamella forming systems confined in circular annular pore induces a few perforated
holes into the lamella morphology. Grain boundaries are induced into the Y-shape, U-
shape, W-shape, V-shape and star shape in the circular annular pores. The lamella
confined in the circular annular pore show straight lamellae strips and curved lamella. In
(a) (b) (c) (d) (e)
(f) (g) (h)
42
the narrow pore size system d = 1, shown in Figure 4.1(a) there are three lamellae on
left half of the pore system, wrapping around (concentric lamellae) the pore system. In
the same pore system, on the upper half of the pore system, there are six straight
lamellae strips normal to the circumference of the boundaries of the pore system.
Between the parallel strips and concentric lamellae, there is one perforated hole induced
in the pore system. Concentric alternating lamellae were conformed due to the curvature
effect in the circular annular pore geometry. In the pore size d = 2, as shown in Figure
4.1(b) a major part of concentric lamellae that was formed in previously sized system
deformed into U-shape and Y-shape lamellae due to the decrease in the curvature.
Furthermore, parallel strips normal to the circumference also disappeared in the circular
annular pore. In this pore size system, five perforated holes adjacent to each other are
displayed in the first quadrant of the pore system shown in Figure 4.1(b) by dashed
lines. Increasing the pore size of the system d = 3 these perforated holes disappeared
and replaced by the curved lamellae which are normal to the circumference of exterior
as well as the interior circular boundary of the pore system as shown in Figure 4.1(c) by
dashed lines. If we compare the system Figure 4.1(b) and Figure 4.1(c) left half of the
pore system then we can observe that concentric lamellae have a tendency to deform
into Y-shape and W-shape with respect to the increase in the pore size. This shows that
microdomains are changing orientation, from parallel to the pore to normal to the pore
system with respect to the increase in the pore size. In this pore size grain boundaries
induced in the pore system are in Y-shape or tilted Y-shape star shape and W-shape. For
the pore system d = 4, we observed the four perforated holes adjacent to each other on
the top of the left half of the pore system as we observed similar perforated holes in the
pore system with the pore size d = 2 on the top of right half of the pore system. The rest
of the domains were patterned with nanodomains in the Y-shape, W-shape U-shape and
a few perforated holes. Furthermore, increasing the size of the pore system d = 5 as
43
shown in Figure 4.1(e) we observed the star lamellae with a perforated hole at the centre
of the star are displayed on the top right half of the pore system as shown inside the
dashed lines. There are similar star lamellae connected with near the interior circular
boundary as well in the pore system. The pore system also displays lamellae in the form
of the two arm stars and single arm stars with a perforated hole at the centre of the star.
In the rest of the pore system grain boundaries are induced in the form of Y-shape, W-
shape and U-shape. In the pore size d = 6 again we observed the perforated holes
adjacent to each other but in this size system, there are six arrays of perforated holes
appeared at the bottom of the right half of the pore system and two perforated holes
adjacent to each other displayed at top right half of the pore system as shown in Figure
4.1(f). However, in the rest of the circular annular pore system microdomains were
induced in Y-shape, W-shape, U-shape and star shape with a perforated hole at the
centre of the star. Furthermore increasing the size of the pore system, grain boundaries
are becoming straight strips at the exterior circular boundary of the pore system and
oriented normal to the pore system as shown in Figure 4.1(g) and Figure 4.1(h). Both
systems are enriched with grain boundaries patterned in Y-shape, W-shape, U-shape,
star shape with a perforated hole at the centre of the star. Y-shape, W-shape, U-shape
and star lamellae are induced in the circular annular pore system due to the curvature
effect of the pore geometry. However, the perforated holes are appearing in the circular
pore geometry due the competition between incompatible blocks and entropy.
Furthermore, spiral morphology did not appear in the pore geometry due to the size
effect as a computational domain under investigation is far away from the centre of the
pore system, where the curvature of the pore geometry has a maximum value.
Simulations results obtained with asymmetric volume fractions for lamella forming
systems are stable and consistent with the experimental and computational studies in the
field; however we predicted some novel morphologies such as star lamellae with a
44
perforated hole at the centre of the star and W-shape lamellae which are to be confirmed
by the experimental studies. The results of this study can be compared with the
experimental work [56], the fingerprints lamella achieved in the planar thin films. In the
planar thin films, lamella mostly forms Y-shape and U-shape patterns, however, here
under geometric confinement results predict perforated holes, star shape and W-shape
strip patterns. The results show bent lamella with openings in the direction of outer
circumference, these patterns are consistent with the results obtained for lamella under
cylindrical confinement where results show bent lamellae pointing towards the column
mantle of the pore Figure 3(c) shown in therein [57]. Fingerprints morphologies were
also observed in planar thin films, microdomains were observed perpendicular to the
substrate and coexistence of fingerprint morphology with perforated holes was observed
in Figure 2(a) therein [84].
In the second case, the interior radius of the annular pore systems was fixed at ar = 5.0
and the pore size of systems increased by the exterior radius br of pore systems. The
exterior radius of pore systems were increased for different values br = 7, 8, 9, 11,
hence the film thicknesses under investigation were d= 2, 3, 4, 6.
45
Figure 4.2: Evolution of asymmetric lamellae system on 1 million time steps in the pore system with the
interior radius of pore system was fixed at ar = 5.0 and the exterior radius br of the pore system was
expanded to obtain various pore sizes, pore size of the systems: (a) evolution of nanodomains in the pore
size d = 2, system show alternating lamella strips normal to the circular boundaries, perforated holes and
Y-shaped defects (b) evolution of lamella patterns in the pore system size d = 3 system induces lamella
strips and Y-shaped defects (c) evolution of the asymmetric lamella system in the pore size d = 4 induced
morphologies are Y-shape, U-shape, W-shape and star shape lamellae with a perforated hole at the centre
of the star (d) nanostructure evolution in the pore size d = 5 induces diverse lamella patterns.
The simulation results for the expanded interior radius show that lamellae are induced
normal to pore walls in the pore geometry. In the system size d = 2, shown in Figure
4.2(a) there are lamellae strips normal to the pore walls and there are perforated holes
adjacent to each other found at top inside right half of the pore system. In the pore
system, Y-shape micro domains are deformed into straight strip or U-shape micro
domains due to the size effect in the geometry. Expanding the pore size of the system d
= 3 lamellae straight strips taking back into Y-shape lamellae in the pore geometry as
shown in Figure 4.2(b). Therefore, it can be argued that morphological transition
between straight strips lamellae and Y-shape lamellae are due the size effect of the pore
geometry. In this system size no perfect perforated holes appeared in the pore system.
Furthermore, expanding the pore system d = 4 we can observe Y-shape, U-shape, W-
shape and star shape lamellae with a perforated hole at the centre of the star appearing
in the pore geometry as shown in Figure 4.2(c). As we go further by expanding the pore
size of the system d = 6 we can observe diversity of lamellae morphological patterns in
the form of Y-shape, U-shape and star lamellae with single arm, double arm, triple arm,
and 4 armed star with perforated hole at centre of the star as shown in Figure 4.2(d).
(a) (b) (c) (d)
46
The simulation results show that the narrow pore sized system induces the parallel strips
of grain boundaries whereas in the expanded pore system these parallel grain boundaries
transform into Y-shape grain boundaries due to the size effect of the pore geometry.
Furthermore, the grain boundaries in the expanded pore system were induced normal to
the pore system.
In the third case, we expanded the pore system furthermore by increasing interior radius
of the annular pore system by keeping it fixed at ar = 7.0 and pore size of the system
was enlarged by increasing exterior radius br of the pore systems. The exterior radius of
the pore systems was increased by expanding the exterior radius of the pore systems for
different sizes br = 9, 10, hence film thicknesses under investigation was d = 2, 3.
Figure 4.3: The asymmetric lamellae system in the expanded pore size with the interior radius of pore
system was fixed at ar = 7: (a) the system size d = 2, system induces alternating parallel lamella strips
normal to the circular boundaries, Y-shaped, W-shaped and star shaped patterns (b) the system size d = 3,
induced patterns including a few alternating lamella strips normal to the circular boundaries, Y-shaped,
W-shaped and star shaped defects.
The simulation results with expanded pore system show parallel lamellae strips normal
to the pore system. In the pore size d = 2, system shows more parallel strips of lamellae
normal to the pore geometry system. Including parallel strips there are Y-shape and W-
shape and star lamellae with perforated hole at the centre of the star also observed in the
pore system as shown in Figure 4.3(a). Increasing the pore size of the pore system d =
3, parallel lamellae strips normal to the system taking the form of Y-shape boundary
grains but there are still few parallel strips normal to the pore system in the pore
geometry as shown in Figure 4.3(b). While rest of the domain in the pore geometry,
(a) (b)
47
patterned into W-shape and double arm star, triple arm star lamellae with perforated
hole at the centre of the star.
II. Asymmetric lamella system with interfacial circular walls
The simulation results for diblock copolymer lamellae forming system with asymmetric
volume fractions in circular annular nano pores system were obtained by applying
preferential attractive walls for the A monomer as well as for the B monomer. In the
circular annular pore geometry, simulation results are obtained by CDS parameter
system for asymmetric lamellae system. Simulation parameters are taken same as in
previous sections for the pore geometry. Periodic boundary conditions were imposed on
the angular coordinates, whereas reflective boundary conditions were imposed on the
radial coordinate in the pore systems. The symmetric boundary conditions were applied
for confining polymer system between two circular walls which carry equal preference
for one of the block of the polymer system throughout the surface. All the simulations
were carried out on 1 million time steps for the pore system. The parameter denotes
the interaction strength between the one of the segment of the polymer system and
confining walls of the pore system.
In the first case of one dimensional confinement, boundary conditions were imposed on
radial axis parallel to the pore system and attractive circular boundary wall with affinity
of A monomer, applied with strength of interaction for various values. The interior
radius of the pore system was fixed at ar = 3, whereas the pore system was expanded by
increasing the exterior radius br of the pore system. In this analysis d denotes the pore
size of the system which shows the difference between the exterior radius of the pore
systems and the interior radius of the pore systems.
48
Figure 4.4: Asymmetric lamellae system under one dimensional confinement obtained on 1 million time
steps, the interior radius of the pore system was fixed at ar =3.0 and the pore sizes of systems are (a) d =
1, pore system induces three alternating concentric lamella rings (b) d = 2.5, pore system induces seven
alternating concentric lamella rings (c) d = 3, system induced concentric lamella and strip lamella (d) d =
4, system induced concentric lamella and dislocations (e) d = 6, system patterned with mixed lamella
morphologies.
The simulation results in the annular circular pore system under geometric confinement
show that the confinement influences the microdomains into concentric lamellae
parallel to the pore system. In the pore size d = 1 with weak interaction applied 2.0
for the pore system as shown in Figure 4.4(a) the system shows perfect concentric
parallel to the pore lamellae in the pore geometry. There is only one circular lamella
with two boundary lamellae appeared in the pore geometry system. The concentric
lamellae parallel to the pore system induced in the pore geometry due to the interplay of
both curvature and confinement effect. The pore system size d = 2 does not show
concentric circular lamellae in the pore geometry not shown here. Furthermore, in the
expanded pore system d = 2.5, perfect concentric circular lamellae were induced in the
pore system as shown in Figure 4.4(b). Between the confining circular walls, it can be
observed that there are seven alternating lamellae induced in circular ring shape in the
pore geometry. As moving further, by expanding the size of the pore system d = 3,
2.0 4.0 8.0 2.0 2.0
2.0 2.0
(c) (a) (b)
(d) (e)
49
system shows that lamellae are tending to wrapping around the centre of the pore
system, however in these sizes system perfect concentric circular lamellae are not
induced as shown in Figure 4.4(c). For this system interaction strength between A
segment of the polymer system and boundary walls were increased from 2.0 to
4.0 and 8.0 , the system shows Y-shape lamellae are converting into concentric
circular lamellae gradually with respect to the size of the system, the system is shown in
three wheels in Figure 4.4(c). For the system size d = 4 as shown in Figure 4.4(d), it can
be observe the Y-shape lamellae are being converting into the concentric lamellae in the
pore system. However, the system with size d = 6 shows mixed orientation of the
lamellae and diverse patterns which shows that in the large systems confining effect is
minimal.
For the similar pore system, simulation results are also obtained by putting interacting
circular walls around the pore system having affinity with B segment of the polymer
system in the pore geometry. The pore system for having affinity of circular walls to B
segment of the polymer system is similar as was designed for counterpart segment A of
the polymer system in the pore geometry.
Figure 4.5: Evolution of asymmetric lamellae system on 1 million time steps under geometric
confinement, interacting circular walls has affinity with the B segment of polymer system, interaction
strength was applied 2.0 , the interior radius of the pore system was fixed at ar = 3, whereas system
was expanded by the exterior radius of the pore system and pore system size are (a) d = 1, system
patterned into concentric lamella (b) d = 2.5, system induces concentric lamella with very few
dislocations (c) d = 3, concentric lamella, perforated holes and dislocations (d) d = 4, mixed patterns.
The pore system confined by putting affinity of the circular walls for B segment of the
polymer system is more effective than the counterpart A segment in the pore geometry.
(a) (b) (c) (d)
50
Figure 4.5 shows the results obtained by confining polymer system by circular walls
around the pore geometry having affinity for B segment of the polymer system. The
pore system size d = 1 shown in Figure 4.5(a) show single perfect circular lamella of B
segment of the polymer system enfolded around the pore. In the second system with
pore size d = 2.5 there are five concentric alternating circular lamellae system induced
in the pore system with defects at two points where circular lamellae were not properly
connected to each other in the pore geometry as shown in Figure 4.5(b). The pore
system having size of the pore d = 3 also shows two concentric perfect circular rings of
lamellae and inside the rings mainly lamellae are circular parallel to the pore system as
shown in Figure 4.5(c). Similarly the system with pore size d = 4 shown in Figure
4.5(d) also shows similar behaviour of the two concentric perfect circular rings inside
the rings lamellae with mixed patterns found in the pore system. The simulation results
predict a confinement effect more effective with having affinity to the circular walls for
B segment of the polymer system in the pore geometry as compared to the counterpart
A segment of the polymer system.
For the expanded pore system with interior radius set fixed at ar = 5 and exterior radius
of system increased to obtained various sizes of the pore system, simulation results have
been obtained by preferential affinity of the surface between circular walls and A
segment of the polymer system. One dimensional confinement is imposed on the radial
coordinate of the pore system, where A segment have preferential affinity to the circular
walls around the pore system. The interior radius of the pore system was kept fixed at
ar = 5 whereas the pore systems were expanded by increasing the exterior radius br of
the pore systems. Here d denotes the pore size of the system which is the difference of
the exterior and the interior radii of the pore systems.
51
Figure 4.6: Asymmetric lamellae system under one dimensional confinement into expanded pore system
with the interior radius was fixed at ar = 5 and the pore size of the systems are (a) d = 2.5 (b) d = 3 (c) d
= 4.
The simulation results under geometric confinement in the expanded pore systems with
interior radius fixed at ar = 5 show parallel concentric circular lamellae and Y-shape
lamellae in the pore systems. In the system size d = 2.5 shown in Figure 4.6(a) with
interaction strength 2.0 system shows concentric circular lamellae parallel to the
pore system and a few packs of perforated holes, however system with increased
interaction strength 4.0 most of the packs of perforated holes disappeared and pore
system induces circular lamellae parallel to the pore system. The pore system with size
d = 3 shown in Figure 4.6(b) shows mostly concentric circular lamellae parallel to the
system under geometric confinement subjected to interaction strength applied 2.0 .
However, in this system there are still lamellae normal to the pore system with a few
perforated holes in the pore system. The pore system with size d = 4 shows mixed
lamellae system induced in the Y-shape parallel to the pore system and lamellae normal
to the pore system as shown in Figure 4.6(c).
For the similar pore system, pore walls activated for having affinity to B segment of the
polymer system in the pore geometry. The strength of interaction between the polymer
segment B and the circular walls are applied 2.0 .
2.0 2.0 2.0 4.0
(a) (b) (c)
52
Figure 4.7: Asymmetric lamellae system under geometric confinement, circular walls have affinity to the
B segment of the polymer system, the interior radius of the pore systems was fixed at ar = 5, and the pore
sizes are (a) d = 2 (b) d = 2.5 (c) d = 3 (d) d = 4.
The simulation results obtained with changing preference of the interacting walls for B
segment of the polymers system show concentric circular rings of the grain boundaries.
There are three concentric perfect circular rings of micro domains apart from circular
boundary walls in the system having size d = 2 as shown in Figure 4.7(a). The pore
system having pore size d = 2.5 also shows three concentric circular rings of boundary
grains induced in the pore geometry as shown in Figure 4.7(b). Both the systems show
same number of micro domains due the size effect of the polar grid system. It can easily
be observed that lamellae in the pore system shown in Figure 4.7(a) are thicker than the
system shown in Figure 4.7(a). The similar structure of concentric alternating lamellae
can be observed in experimental study [15] there in Figure 1(A). This type of
morphology is also called dartboard morphology. Using moderate surface interaction
Sevink et al computationally obtained dartboard morphologies in various pore sizes [17]
Figure 4 therein but these observed alternating concentric were not perfectly circular.
These onion-like concentric alternating layered lamella were also obtained by the
preparing lamella in the spherical pores [85], see Figure 1 therein.
The pore system with size d = 3 as shown in Figure 4.7(c) shows concentric circular
lamellae with a few packs of the perforated holes displayed inside the pore system. The
pore system having size of the system d = 4 displays in one half of the pore system
concentric circular lamellae and other half of the circle mixed behaviour of lamellae in
the pore system as shown in Figure 4.7(d). The simulation results show that changing
(a) (b) (c) (d)
53
preference from A segment to B segment of the polymer system influence more to form
periodic patterns of concentric circular lamellae ring in the pore geometry.
Furthermore, expanding the pore system through the interior radius of the pore system
by keeping it fixed at ar = 7 and the exterior radius of the system was increased to
obtained various sizes of the pore geometry, simulation results were obtained in the pore
system by preferential affinity of the surface between circular walls and A segment of
the polymer system. One dimensional confinement is imposed with symmetric
boundary conditions on the radial coordinate of the pore system, where A segment have
preferential affinity to the circular walls around the pore system. The interior radius of
the pore systems was kept fixed at ar = 7, whereas the pore systems were expanded by
increasing the exterior radius br of the pore system. Here d denotes the pore size of the
system which is the difference of the exterior and the interior radii of the pore system.
Figure 4.8: Asymmetric lamellae system under geometric confinement, the interaction strength was
applied 2.0 , the expanded pore system with the interior radius of the pores were fixed at ar = 7 and
pore sizes of the systems are (a) d = 2.5 (b) d = 3.5.
The simulations results under geometric 1-D confinement into expanded pore system
show perforated holes and lamellae normal to the pore system. The interior radius of the
pore systems were fixed at ar = 7, whereas the exterior radius of the pore systems were
expanded as br = 9.5, 10.5 hence results were obtained in the pore sizes d = 2.5, 3.5.
Due to the large system impact of the curvature in the pore geometry on microdomains
is nominal. The pore system under one dimensional confinements are shown in Figure
4.8(a) with the size of the system d = 2.5 and in Figure 4.8(b) with size d = 3.5. The
(a) (b)
54
pore system having pore size d = 2.5 induces three alternating circular lamellae with
defects at two sections of the pore system where circular lamellae are not properly
connected to each other. However, the pore system with size d = 3.5 did not show
perfect circular lamellae in the pore geometry. This system size show mixed behaviour
of lamellae in the pore geometry where lamellae are in parallel to the pore, normal to
the pore and perforated holes are induced in the pore system. This shows that due to the
size effect there is nominal inclination of grain boundaries to reorient themselves into
parallel to the pore system.
For the similar pore systems simulation results are also obtained by changing preference
to the circular walls from A segment to B segment of the polymer system in the pore
geometry. The interaction strength between the circular walls and the B segment of the
polymer system is set at 2.0 in the pore geometry.
Figure 4.9: Lamellae system under geometric confinement, circular walls have affinity to the B segment
of the polymer system, the interior radius of the pore system was fixed at ar = 7, the pore size expanded
by the exterior radius of the pore system, pore sizes are (a) d = 2.5 (b) d = 3.5.
The pore system with changing preference for monomer B shows lamellae in the form
of circular ring in the pore geometry. For the system having pore size d = 2.5 displays
three perfect circular rings of alternating lamellae induced in the pore system as shown
in Figure 4.9(a). The system with pore size d = 3.5 shows two concentric circular rings
of lamellae and inside the rings there is mixed behaviour of lamellae system where
some packs of circular lamellae and some packs of lamellae are normal to the pore
system as shown in Figure 4.9(b).
(a) (b)
55
The concentric lamella (onion-like) structure are obtained here are very much consistent
with the results obtained for lamella system confined in cylindrical pore by the
experimental work [57] shown in Figure 3(a) therein.
4.2.2. Symmetric lamella forming system confined in circular annular pores
The simulation results for diblock copolymer lamellae morphology for symmetric
volume fractions in circular annular pores with variations in diameter obtained on one
million time steps by using CDS method employed in polar coordinate system. Volume
fractions play a vital role in nano fabrications of block copolymer systems. The
simulation results were obtained with the symmetric volume fraction of CDS parameter
system to investigate the influence of volume ratio on diblock copolymer lamellae
forming system in the pore geometry. For the simulation results of lamellae system with
symmetric volume fractions CDS parameters u and D also changed along with f.
Following table (3) shows CDS parameters system used in the simulations; however the
simulation parameters used were same as in previous section.
Table 3: CDS parameters for symmetric lamella system
f u v B D A
0.36 0.50 0.50 2.30 0.02 0.50 1.50
The interior radius of pore system was fixed at ar = 3, however, the exterior radius of
pore systems was varied to expand the pore size of the systems as br = 5,7,9, hence
domain sizes of pore under consideration were d=2, 4, 6. The Snapshots of CDS
simulation results are shown in following Figure 4.10.
56
Figure 4.10: Symmetric lamellae forming system confined in circular annular pores with the interior
radius of the pore system was fixed at ar = 3.0 and the pore sizes are (a) 2 (b) 4 (c) 6.
The computational results for diblock copolymer lamellae forming system in circular
annular nano pores with using equal volume fractions of both monomers were obtained.
The results show more defect lamellae as compared to previous sections results obtained
with asymmetric volume fractions in annular circular pore system. In the simulation
results shown in Figure 4.10, it can be observed that they are more diagonal as
compared to the simulation results obtained with non-symmetric volume fractions
parameter system. It can also be noted here that they are discontinuous as compared to
the asymmetric volume fraction case. Similarly, the like behaviour of lamellae normal
to the circumference of the exterior circle can be observed in symmetric case but in this
case normal lamellae to circumference of exterior circular boundary of pore are small in
length and discontinuous. In the middle of the domain of hollow disk lamellae shows
mix behaviour like normal to the pore system and parallel to to the pore system and
shows more diagonal shapes. Therefore the important thing observed by comparing both
results of asymmetric case and asymmetric case is that asymmetric lamellae are more
curved and continuous whereas in symmetric case lamellae are more diagonal and
discontinuous.
For the similar pore system, simulation results were obtained by one dimensional
geometric confinement by giving preference to the circular walls for A segment of the
(a) (b) (c)
57
polymer system. The interaction strength between circular walls and A segment of the
polymer system was set at 2.0 .
Figure 4.11: Symmetric lamellae system under geometric confinement with the interior radius of the pore
system was fixed at ar = 3.0 and pore sizes of the system are (a) d = 3 (b) d = 6.
Symmetric lamellae system under one dimensional geometric confinement in annular
pore system shows grain boundaries in mixed orientation between confined walls in the
pore geometry. The pore system having size d = 3 shows that few micro domains are
parallel to the circular wall and others are normal to the circular walls in the pore
geometry as shown in Figure 4.11(a). The pore system with size d = 6 shown in Figure
4.11(b) also shows mixed behaviour of the boundary grains with few perforated holes
inside the pore system. This shows that symmetric lamellae parameter system in CDS is
not efficient to study lamellae forming system in the pore geometry.
4.2.3. Cylindrical forming system confined in circular annulus pores
Cylindrical morphology of diblock copolymer investigated in circular annular nano
channels without confinement’s walls and with geometric confinements in one
dimension as well as two dimensions. The cylindrical forming system can be achieved
by using CDS parameters system for volume fraction f = 0.40 and temperature
parameter = 0.30. The simulation results were obtained in circular annular pore with
keeping interior radius of the pore system fixed and expanding exterior radius of the
system on various sizes of pore. In the pore geometry simulation results were obtained
on one million time steps.
(a) (b)
58
I. Cylindrical forming system confined in neutral circular annular pores
The computational results for cylinder forming in circular annulus pores were obtained
by keeping the interior radius of the pore fixed and expanding exterior radius of pore
system for various values of radius. The simulation results were obtained on 1 million
time steps in annular circular pore system. The CDS Simulations were performed using
CDS parameters for cylindrical forming system shown in following table (4).
Table 4: CDS parameter system for cylindrical forming system
f u v B D A
0.30 0.40 0.50 1.50 0.02 0.50 1.50
The simulation results were obtained in a pore system which is contained by concentric
circles of radius ar and br , where ar represents interior radius of the pore system and br
represents the exterior radius of the pore system. The interior radius of pore systems
were fixed at ar = 3.0 and the pore systems were extended by increasing the exterior
radius of pore systems as br = 4, 5, 6, 7, 9. The size of the pore system is represented
here by d = br - ar . The simulation results were obtained with various pore sizes are
shown in Figure 4.12.
Figure 4.12: Cylindrical system in the pore geometry with pore sizes (a) d = 1 (b) d = 2 (c) d = 3 (e) d =
4 (f) d = 6
(a) (b) (c)
(d) (e)
59
Cylindrical system in the pore geometry shows that microdomains make alignment on
the logarithmic spiral lines in the pore system. Microdomains also show the hexagonal
packing arrangements in the pore geometry. In Figure 4.12(a) the system is shown with
pore size d = 1 cylinders lie on the logarithmic spiral lines where spiral line are shown
by dashed lines. In this system spiral rays are narrowed or spiral lines have short
distances from the origin. The pore system with size d = 2 also shows that the cylinders
are aligned on the logarithmic spiral lines where spiral lines are shown by the dashed
lines as shown in Figure 4.12(b). In this system spiral rays are bit wider or have a more
distance from the centre of the pore system as compared to previous sized pore system.
The pore system having pore size d = 3 as in Figure 4.12(c) shows micro domains lie on
the spiral lines having almost same spiral rays distance from the origin of the pore
system. However, the pore system with size of the pore d = 4 shows cylinders on the
spiral lines having spiral rays wider due the size effect of the pore geometry.
Furthermore, pore system with size d = 6 shows micro domains on the spiral lines with
spiral rays very much wider due the size effect of the pore geometry as shown in Figure
4.12(d). The cylinders in the pore system are also hexagonally packed as shown in
Figure 4.12(d) along with zoomed snapshot of the image. This shows that in the pore
geometry cylinders align on the spiral rays and spiral arms becomes wider with respect
to increase in the pore size of the system. The experimental study shows that the
frustration caused by the confinement distorts the natural packing of hexagonal
arrangements of the microdomains [86, 87].
The pore system is also expended by the increasing interior radius of the pore system at
two fixed values ar = 5 and ar = 7 and the exterior radius of the pore system was
increased at different values to expand the pore system. In both pore systems we
obtained similar patterns of the cylindrical forming system in the pore geometry. The
cylindrical forming systems were observed packed along the spiral lines in the pore
60
geometry due to curvature influence. Hexagonal packing of the micro domains was also
observed inside the pore geometry.
II. Cylindrical system with interfacial circular walls
Diblock copolymer cylindrical forming system was studied in circular annulus pore
system using preferential attractive circular walls having interaction strength 2.0
between the walls and one of the segments of the polymer system. In one dimensional
confinement attractive circular wall interaction with one of the segments of the polymer
system applied in radial r direction in annular pore system. The simulation results were
obtained in a pore system by keeping fixed the interior radius of the pore system ar = 3
and pore size was expanded by increasing exterior radius br of the pore system. The
parameter d = br - ar represents the pore size of the pore geometry.
In the first case, attractive wall for interaction with monomer A was applied in one
dimension in the pore system.
Figure 4.13: Cylindrical systems confined under interfacial circular walls having affinity to A block and
the pore size of systems are (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.
The simulation results obtained in one dimensional circular confinement show that the
surface preference influences the micro domains to form circular alignment in
concentric circles around the centre of the pore system. Surface affinity to A segment of
the polymer system distort the spiral alignment of the micro domains pack them in the
concentric circles in the pore geometry. However, there are still micro domains packed
on the rough spiral lines inside the pore system. Micro domains which had been
(a) (c) (b) (d)
61
distorted from both spiral and circular packing were due to the packing frustration
caused by interplay of both curvature and confinement effects. In pore system having
pore size d = 2 as shown in Figure 4.13(a) the cylinders were packed in three concentric
circles in the pore system. The pore system with size of the pore d = 3, micro domains
were packed in four concentric circle in the pore geometry. The pore size d = 4,
contains cylinders packed in six concentric circles in the pore system. In this sized
system micro domains are not well packed in the in circular lines due the size effect.
The pore size d = 6, contains eight circular lines of the micro domains in the pore
system. In this sized system micro domains are also not well packed in the circular lines
due to the size effect.
The simulation results for diblock copolymer cylindrical system with preferential
attractive circular wall with interaction between circular walls and monomer B were
obtained in circular annular pore system. Interaction strength 6.0 applied between
the circular walls of the pore system and B segment of the polymer system in the pore
geometry. Similarly, the simulation results were obtained in a pore system whose
interior radius ar = 3 of the pore system was fixed and the pore size was extended by
exterior radius br of the pore system, where d defined the pore size of the pore system.
Figure 4.14: Cylindrical systems with interfacial affinity to B block and the pore size of the systems are
(a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.0.
The cylindrical forming system under geometric confinement with circular walls having
affinity with B monomer shows similar patterns as was in the case when walls had
affinity with A monomer in the pore geometry. Microdomains are packed along the
(a) (b) (c) (d)
62
concentric circles inside the pore system. The pore system of size d = 2 shows micro
domains packed into two concentric circles in the pore geometry as shown in Figure
4.14(a). However, similar sized system pore walls having affinity with the A monomer
showed micro domains are packed into three concentric circles in the pore system which
is due to the incompatibility of segments of the polymer system. Incompatibility of both
segments squeeze down the micro domains because the A segment remains outside of
the micro domains and when the B segment was placed on the circular walls. The pore
system d = 3 shows cylinders are packed in three concentric circles in the pore system
as shown in Figure 4.14(b). Similarly the pore system d = 4 shows micro domains are
aligned in five concentric circles in the pore geometry as shown in Figure 4.14(c).
Finally the pore system d = 6 shows micro domains are packed in seven concentric
circles in the pore system as shown in Figure 4.14(d).Simulation results show that the
spiral packing of micro domains is influenced by the curvature effect whereas the micro
domains packed into concentric circles due the confinement effect in the pore geometry.
4.2.4. Spherical forming system confined in circular annular pores
AB Diblock copolymer spherical forming systems is explored in circular annular pore
with various sizes of pore system. Spherical morphology is investigated without
interacting walls in the pore geometry as well as with interacting walls (circular walls)
with one of the segment of the polymer system in the pore system. Refinement of CDS
parameters were also carried for better choice of phenomenological constants.
I. Sphere forming system confined in the neutral circular annular pores
The Simulation results were obtained for spherical forming system in circular annulus
pore systems with various pore sizes, by keeping interior radius of pore system fixed at
ar = 3 and pore size was expanded by increasing exterior radius of the pore system.
Simulation results were obtained on 1 million time steps. In the pore geometry
simulation results were obtained without confining circular walls. The Simulations
63
results for the sphere forming system were obtained by initializing the order parameters
in the range Ψ = ±0.5 in the annular pore system. The following table 5 shows CDS
parameter system for sphere system with higher temperature entered in the simulation.
Table 5: CDS parameters on higher temperature for sphere system
f u v B D A
0.20 0.40 0.38 2.30 0.01 0.50 1.50
The pore system is contained between two concentric circles of radii ar and br , where
ar represents the interior radius of the pore system and br represents the exterior radius
of the pore system. The interior radius of the pore systems ar = 3 was fixed. The size of
the pore systems was increased by increasing the exterior radius of pore systems as br =
5, 6, 7, 9. The pore sizes of the circular annular system is denoted by d = br - ar which
shows the difference between the interior radius and the exterior radius of the pore
systems.
Figure 4.15: Sphere system in the circular annular pore geometry and the pore sizes are (a) d = 2 (b) d =
3 (c) d = 4 (d) d = 6.
The simulation results for sphere forming system show various arrangements in the pore
system. Figure 4.15(a) shows the system with size d = 2, micro domains are packed
along the spiral lines in the pore system. Figure 4.15(b) shows the system size d = 3,
which also show that spheres are induced in the system along the spiral rays and spiral
rays are very wide in the pore system. This shows the size effect of the system due to
(a) (b) (c) (d)
64
the curvature in the pore geometry. However, the pore system d = 4, show
rhombohedral arrangements of the sphere in the pore system as shown in Figure 4.15(c)
by dashed lines. Furthermore, the pore system d = 6 show spheres are packed along the
hyperbolic lines as well as hexagonal arrangements in the pore system as shown in
Figure 4.15(d) by dashed lines.
The CDS parameters were investigated by the different choice of values for
phenomenological constants including the high temperature parameter value. Following
table shows the CDS parameters chosen for sphere forming system in circular annular
pore system. The simulation results were obtained in similar pore system as above in
which interior radius of the pore systems were fixed at ar = 3 and pore size of the
systems was extended by increasing the exterior radius of the pore systems.
Table 4 Modified CDS parameters for sphere forming system
f u v B D A
0.25 0.40 0.50 1.5 0.02 0.50 1.50
Figure 4.16: Sphere forming system with modified CDS parameter system and pore system sizes are (a) d
= 2 (b) d = 3 (c) d = 4 (d) d = 6.
The simulation results were obtained with the modified CDS parameters for sphere
forming system in circular annular pore system. In the pore system with modified CDS
parameter system sphere system is more ordered as compared to the previous Parameter
system. Sphere systems are induced in hexagonal arrangement in the pore systems. The
pore system d = 2 show spheres are induced along the spiral lines in the pore geometry
(a) (b) (c) (d)
65
as shown in Figure 4.16(a). In this sized system, hexagonal arrangements of the sphere
forming system in the pore geometry can be observed. The pore system d = 3 also
shows spheres are packed along the spiral lines and hexagonal arrangements of the
sphere system in the pore geometry as shown in Figure 4.16(b). As pore size increase
the sphere system induced along the spiral lines vanish but the sphere system remains in
the hexagonal packing in the pore, the system pore system d = 4 is shown in Figure
4.16(c). The pore system d = 6 also shows hexagonal arrangements of the sphere
forming system in the pore geometry as shown in the Figure 4.16(d).
II. Sphere system with interfacial circular walls
The simulation results were obtained for the sphere forming system in circular annulus
nano pores using one dimensional confinement by placing affinity of the circular walls
to the one of the segments of the polymer system. One dimensional confinement
boundary conditions were applied by interior and exterior circular walls of pore system
with interaction strength 2.0 . The CDS parameters system with low temperature as
shown in table 5 is used for sphere forming system. Similar pore system was used in the
simulations as was in the previous section. In the first case, the simulation results were
obtained with confining interacting circular walls having affinity to the A monomer of
the polymer system in the pore geometry.
Figure 4.17: sphere forming system with system subject to the affinity of interacting walls to A block and
system size (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.
(a) (b) (c) (d)
66
The simulation results were obtained under geometric confinement in the pore geometry
and show alignment in the concentric circles and rhombus arrangements. The pore
system d = 2 show sphere aligned along two concentric circles in the pore system as
shown in Figure 4.17(a). Similarly the pore system d = 3, show micro domains aligned
along the three concentric circles in the pore system as shown in Figure 4.17(b).
However, the pore system d = 4 show sphere packing along five concentric circles in
the pore system as shown by Figure 4.17(c). Similarly the pore system d = 6 show
micro domains are packed along seven concentric circles in the pore system as shown
by Figure 4.17(d). In all system sizes spheres are also arranged in rhombus packing as
well which was predicted in the unconfined system as well in previous section.
The simulation results were also obtained for sphere forming system under geometric
confinement where circular walls have affinity to the B segment of the polymer system
in the pore geometry. The interaction between the circular walls and the B segment of
the polymer system was applied 2.0 in the pore system.
Figure 4.18: Sphere system with preferential attractive wall for monomer B and the pore system size are
(a) d = 2 (b) d = 3.
The simulation results show that the B block is not properly forming the circular walls
around the pore system due to the low temperature and minority segment of the polymer
system as shown in Figure 4.18. Locations of the circular walls spheres are induced in
the system that shows that there is minimal impact of confinement on the polymer
system in the pore geometry. The simulation results were also obtained by increasing
the interaction strength between the circular walls and the B segment of the polymer
(a) (b)
67
system but proper circular walls could not arise in the form of B segment around the
pore system.
In the pore system simulation results were also obtained by modified CDS parameter
system with high temperature, CDS parameters are shown in table 8. Similar pore
system is used in the simulation results where interior radius of the pore system is fixed
at ar = 3 and pore size expanded by the exterior radius br of the pore system.
In the first case interacting circular walls applied around the pore system having affinity
to A segments of the polymer system in the pore geometry. Interaction strength between
the circular walls and A segment of the polymer system set at 2.0 in the pore
geometry.
Figure 4.19: Sphere forming system with modified CDS parameter system subject to the affinity of
interacting walls to A block and system sizes are (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.
The sphere forming system obtained by modified CDS parameter system under
confinement having affinity of circular walls to the A segment of the polymer system
show micro domains are packed along perfect concentric circles in the pore geometry.
In this system micro domains are squeezed down under geometric confinement in the
pore geometry. The pore system d = 2 show spheres packed in three concentric circles
as shown in Figure 4.19(a). Due to the squeezing down the micro domains we observed
one extra circular line of the sphere system as compared to previous confined results
with CDS parameter system with low temperature in the pore system. The pore system
d = 3, shows micro domains are packed in four concentric circles in the pore system as
(a) (b) (c) (d)
68
shown by the Figure 4.19(b). Similarly the pore system d = 4, displays five rings
packed with sphere system in the pore geometry as shown in Figure 4.19(c). The pore
system d = 6, shows spheres are induced into eight rings of concentric circles in the
pore geometry. This shows that under geometric confinement sphere system induces
into the rings in the pore geometry. It can be easily observed that spheres are also
arranged in hexagonal packing in the pore geometry under geometric confinements.
The simulation results were also obtained for sphere forming system by modified CDS
parameter system using interacting walls having the affinity to the B segment of the
polymer system in the pore geometry. The interaction strength between the circular
walls around the pore system and the B segment of the polymer system applied is
6.0 in the pore system. The pore system is similar with the interior radius of the
system is fixed at ar = 3 and the pore size is extended by the increasing the exterior
radius br of the pore systems.
Figure 4.20: Sphere forming system with modified CDS parameter system subject to the affinity of
interacting walls to A block and system size (a) d = 2 (b) d = 3 (c) d = 4 (d) d = 6.
The simulations results were obtained by placing interacting walls having affinity to the
B monomer of the polymer system in the pore system. The simulation results show
circular alignment in the concentric circles around the pore system in the pore geometry
as shown in Figure 4.20. Microdomains in this case squeezed down their size due the
frustration caused by the preference of the minority segment on the boundary walls in
the pore system. Microdomains are also packed in rhombus arrangements in the pore
system. Figure 4.20(a) show the pore system with size d = 2 micro domains are induced
(a) (b) (c) (d)
69
into two concentric circles between the circular walls occupied by the B segment of the
polymer system in the pore geometry. The pore system with size d = 3, in Figure
4.20(b) show spheres are packed in three concentric circles in the pore geometry
surrounded by the circular walls occupied by the B segment of the polymer system.
Similarly the pore system having size of the pore d = 4, show microdomains are
induced into the four concentric circles in the pore system surrounded by the B segment
of the polymer system in the pore geometry as shown in Figure 4.20(c). The pore size
system d = 6, shows spheres packed into seven concentric circles surrounded by the
circular walls containing B segment of the polymer system in the pore geometry. The
experimental studies [86, 87] show that the confinement distorts the natural packing
arrangements of the microdomains.
4.3. Summary
Using CDS method employed in polar coordinate system we obtained novel
nanostructures in the circular pore geometry for lamella, cylinder and sphere forming
systems. Asymmetric lamella forming system in the neutral film interfaces shows
perpendicular to the exterior circumference and parallel to the interior circumference
tendency in the circular pore system. Grain boundaries confined in the circular pore
geometry show various formations of lamella including Y-shape, U-shape, W-shape, V-
shape and T-junctions. Fingerprint morphology confined in the neutral circular pore
with various pore thicknesses induced clusters of perforated holes, isolated perforated
holes and star lamella with perforated hole at the centre of star. Close to the exterior
pore boundary, lamellae conforms concentric parabolic patterns having openings in the
direction of outer boundary, however for larger system sizes these parabolic patterns
evolve into the parallel lamella strips normal to the outer circular boundary. The pore
system with large the interior radii shows parallel strips lamellae normal to the circular
70
pore boundaries. The interplay of both interfacial circular walls and curvature influence
induces lamella into the concentric circular rings in the circular pore geometry. In the
presence of interfacial circular walls, concentric alternating lamella (onion-like)
nanostructure were obtained in the circular pores which are strongly consistent with
experimental results. Results show that under the influence of curvature and interfacial
circular walls, the microdomains tend to form concentric lamella in the pore geometry.
Symmetric lamella system induces defective nanostructures in the circular pore
geometry.
Cylindrical forming system in the neutral circular annular pore system shows, the novel
packing arrangements along the spiral lines. However, in the larger pore systems spiral
packing arrangements alters and system regains in the classical hexagonal packing
arrangements in the circular pore system. While, in the presence of interacting circular
walls microdomains induces packing arrangements in the concentric circular rings in the
pore geometry.
Similarly, sphere forming systems show, packing arrangements along the spiral lines,
parabolic lines with opening along the outer boundary and hexagonal packing
arrangements in the pore system. A new set of CDS parameters for sphere forming
system were also introduced in the study. Results obtained with modified CDS
parameters show that small system size induces sphere along the spiral curve, while
large circular pore system induces sphere in the classical hexagonal packing
arrangements. In the presence of interfacial circular walls, the sphere system with
modified CDS parameters was squeezed down in size and shape. The sphere system
under geometric confinement of annular circular pores with the interfacial circular walls
shows, packing arrangements in the concentric circular rings. The results show that
under the influence of the curvature nanodomains show spiral packing arrangements,
71
while, due to the interplay of both curvature and interfacial circular walls microdomains
forms packing arrangements in the concentric circular rings in the pore geometry.
72
CHAPTER FIVE
5. Block copolymers confined in cylindrical pores
5.1. Introduction
Block copolymer were investigated under the geometric confinements of hollow
cylindrical pores using the CDS method employed in the cylindrical coordinate system.
Lamella, cylinder and sphere forming systems are obtained in the cylindrical pore
geometry. Formations of lamella, cylinder and sphere morphologies were obtained in
cylindrical surfaces with various pore radii and pore lengths. The results were obtained
for block copolymers confined in neutral cylinders pores and with interfacial surfaces.
In the case of interfacial surfaces, symmetric boundary conditions were applied on the
pore interfaces. The interaction strength between the pore walls and the majority
segment of the polymer system is represented by parameter . The results for block
copolymers confined in the cylindrical pore were obtained with various interaction
strengths in the pore geometry. In the case of interfacial cylindrical surfaces results were
also obtained by applying two dimensional interfacial surfaces. Two dimensional
interfacial surfaces were applied by parallel to the pore surface circular walls and
perpendicular to the pore surface covering top and bottom of the cylindrical pores. For
the parallel interfaces boundary conditions were applied on the radial coordinate of the
lattice, while, for perpendicular interfaces boundary conditions were applied on the
length coordinate z of the pore lattice. In the cylindrical pores, results were obtained by
keeping the interior radius ar fixed, while, the pore size was expanded by increasing the
exterior radius br of the pore surface. The results were obtained for block copolymers
confined in cylindrical surfaces by applying periodic boundary conditions on the
73
angular coordinate and pore length coordinate z. In addition, results were obtained by
applying reflective boundary conditions on the radial coordinate r.
5.2. Results and discussions
The AB diblock copolymer lamella forming system, cylindrical forming system and
spherical forming patterns were investigated in hollow cylindrical nano channels using
the CDS method employed in the cylindrical coordinate system. Study and analysis
were carried out for diblock copolymers with neutral surfaces and with confining
interacting walls. The simulation results in cylindrical pore system were obtained under
one-dimensional geometric confinement as well as two-dimensional geometric
confinements are presented. The simulation results were obtained using an initial
random disordered state for order parameter ( ) which was a random number within
the range Ψ = ±1. All the simulations were carried out on 1 million time steps using
CDS method employed in cylindrical coordinate system.
5.2.1. Lamella forming system confined in cylindrical pores
The AB diblock copolymer asymmetric lamella forming system was investigated in
cylindrical nano channels. The CDS simulation results for lamella forming system were
obtained by applying asymmetric volume fraction parameter system. The asymmetric
volume fraction is used with f =0.48 which is different from the case f =0.50 as shown
in results of circular annular pore system in chapter 3 that asymmetric parameter system
works well as compared to symmetric lamellae system in the pore geometry.
I. Lamella system confined in neutral cylindrical pores
In the first case, the simulation results were obtained in a pore system without applying
preferential interacting walls around pore system. The simulation results were obtained
with various pores sizes and pore radii of the pore geometry.
74
Table 5: CDS parameter system for asymmetric lamella forming system
f u v B D A
0.36 0.48 0.38 2.30 0.02 0.70 1.50
In the first case simulations results are obtained for asymmetric lamellae system in the
cylindrical pore system where the interior pore radius was fixed at ar = 3 and the pore
size were varied by the exterior pore radius br . The pore size is defined by d = br - ar ,
where ar represents interior radius of the pore system and br represents the exterior
radius of pore system. The length of the pore system was kept fixed h = 4 and angular
coordinate was varied 20 in all simulation results.
Figure 5.1: Lamella system confined in neutral cylindrical pores, with pore system where interior radius
was fixed at ar = 3 and the pore size (a) d = 0.5 (b) d = 1.5 (c) d = 2.
The simulation results in the pore system with the interior radius fixed at ar = 3 show
helical morphology of lamellae system in the pore geometry. The pore size d = 0.5
system show single helix of patchy lamella in the pore system as shown in Figure
5.1(a). However, the pore system with size d = 1.5, show single helix of flat lamella in
the pore system as shown in Figure 5.1(b). Furthermore increasing the size of the pore
system d = 2.0 we get single helix of lamella with defect in the section where lamellae
becomes parallel to the pore walls as shown by dashed lines in the Figure 5.1(c).
In the second case, the simulation results are obtained in a pore system with interior
pore radius was fixed as ar = 5 and pore size were varied by the exterior pore radius br .
The pore size is defined by d = br - ar , where ar represents the interior radius of the pore
(a) (b) (c)
75
system and br represents the exterior radius of pore system. The length of the pore
system was fixed at h = 4 and angular coordinate was varied 20 in all the
following simulation results.
Figure 5.2: Lamellae system confined in the neutral cylindrical pores, interior radius of pore system fixed
at ar = 5 and pore size (a) d = 0.5 (b) pore size d = 1 (c) d = 1.5 (d) d = 2 (e) d = 2.5 (f) d = 3.
The simulation results of the lamella system confined in neutral surfaces shows
sequences of lamellae sheets flaked in circular alignment mostly perpendicular to the
pore system. In narrow pore system (pore size d = 0.5), long curved strings of lamellae
were found which are parallel to the pore as shown in Figure 5.2(a). Parallel to the pore
patterns of lamellae system forms due to the size effect. Increasing the size of pore
system (d = 1.0) result is shown in Figure 5.2(b) these lamellae strings adapt sheets
structure parallel to the pore system and perpendicular to the pore system. Further
increasing the size of the pore (d = 1.5), most these lamellae sheets became
perpendicular to the pore system. This shows that increase in the pore size of the system
lamellae sheets tends to perpendicular to the pore system. The pore system shows that
micro domains on the interior boundary of the pore system tend to become parallel to
the pore system while rest of the micro domains tend to perpendicular to the pore
system in the pore geometry. The simulation results show that increasing the pore size
of the pore system lamellae tend to become perpendicular to the pore system in the pore
(a) (b) (c)
(d) (e) (f)
76
geometry. The pore system with size d = 3, show perpendicular to the pore lamellae
system in the pore geometry as shown in Figure 5.2(f).
I. Lamella system with interfacial surfaces
Computational results were obtained in the cylindrical pore system with applying
preferential attractive walls around polymer system. The strength of interaction between
circular walls and one of the blocks of polymer system is defined by the parameter in
the pore system. In first case, preferential affinity to the pore surface for one of the
blocks set at 2.0 . Simulation results were obtained by one-dimensional confinement
in radial direction so that polymers are confined between two concentric circular walls
parallel to the pore system.
In the first case, interior radius of the pore system was fixed at ar = 3 and exterior radius
br of the pore system increased to expand the pore system.
Figure 5.3: Lamella system with interfacial surfaces, the pore system with interior radius fixed at ar = 3
and the pore sizes are (a) d = 1.0 (b) d = 1.5.
The simulation results under geometric confinement show concentric lamellae
cylindrical sheets in the pore geometry system. In the pore system d = 1 show single
lamella cylindrical sheet under the influence of the geometric confinement in the pore
system as shown in Figure 5.3(a). However, in the pore system with pore size d = 1.5,
we observed two concentric lamellae sheets in the pore system under the influence of
geometric confinements by circular parallel walls as shown in Figure 5.3(b) along with
interior layer of the pore system.
(a) (b)
77
In the next case, the interior radius of the pore system was fixed at ar = 5 and exterior
radius br of the pore system increased to expand the pore system.
Figure 5.4: Lamella system with interfacial surfaces, interior radius of the pore surface was fixed at
5ar and the pore sizes are (a) d = 1.0 (b) d = 1.5(c) d = 2.5 (d) d = 3.0.
Under geometrical confinement diblock copolymer lamellae forming system in
cylindrical pores displays concentric lamellae cylindrical sheets for preferential affinity
of surface to one of the monomer of the polymers system. It shows that lamellae plates
appeared in the unconfined system shown in previous section results transformed into
cylindrical sheets under geometrical confinement in the pore system. These results show
(a) (b)
(c)
(d)
78
consistency with what we have achieved in simulation results obtained in annular
circular pore system with one-dimensional confinement. Perfect parallel cylindrical
sheets of lamellae system is a results of both curvature and confinements. In the system
with dimensions (d = 1.0) have appeared with single-layer of cylindrical sheet of
lamella parallel to the pore system as shown in Figure 5.4(a). The Pore system with size
and length (d = 1.5), displays two layers of lamellae sheets parallel to the pore system
as shown in Figure 5.4(b) with interior layer of lamellae sheet adjacent to the image. For
pore size and length (d = 2.5), the obtained results show three lamellae layers of
cylindrical sheets parallel to the pore system shown in Figure 5.4(c) with both interior
layers adjacent to the main image. The cylindrical pore system with size and length (d =
3.0), shows four lamellae layers of cylindrical sheets parallel to the pore system as
shown in Figure 5.4(d). In the pore system strong interaction strength 4.0 also
applied between the majority segment of the polymer system and surface walls but in
this case, similar results were observed in the pore geometry. The concentric lamella
(onion-like) structure were obtained here are very much consistent with the results
obtained for lamella system confined in cylindrical pore by the experimental work [57]
shown in Figure 3(a) therein.
Subsequently, results were obtained by applied interfacial surfaces to the majority
segment of the polymer system in z direction covering top and bottom of the cylindrical
pore system through cross sections. The strength of interaction between microdomains
and surface walls was same 2.0 and was applied in perpendicular to the pore
system. In the experimental study of PS-b-PBD lamella confined in the cylindrical pores
concentric cylinder morphology formed within the cylindrical pores, in addition, study
show the number of cylinders depends on the ratio of the pore diameter to the
equilibrium period of the copolymer [87]. The concentric-layered morphology of
lamella system was also obtained in cylindrical pore by the experimental study [15].
79
Figure 5.5: Cylindrical system confined by interfacial surfaces perpendicular to the pore system and the
pore size and pore lengths are (a) d = 1.6 and h = 1.6 (b) d = 2.5 and h = 1.0 (c) d = 2.5 and h = 2.5.
In the system with interacting boundary walls applied in z direction system shows
circular disks perpendicular to the pore system. Lamellae cylindrical sheets parallel to
the pore system in parallel confinement transformed into disks perpendicular to the pore
system in perpendicular confinement due to preferential affinity of the surface changed
from parallel walls to perpendicular walls. In the pore system shown in Figure 5.5(a)
there are two disks in the system one on top of each other. The top view of the system
with top layer and bottom layer of the system also shows adjacent to the image Figure
5.5(a). The obtained disks are not flat from inside and are connected with each other at
four points at the exterior boundary of the pore system. Inside the disks there are thick
Top
view Top
layer
Bottom
layer
(a)
Top
view
Top
layer
Bottom
layer
(b)
(c)
80
layers patterned on the interior surface of disks and the outer surface of disks is flat.
This is due the interaction of polymer blocks with each other. Subsequently squeezing
domain size (d = 2.5 and h = 1) system shown in Figure 5.5(b) two very thin layered
disks appeared in the pore system. The image of the system shown in Figure 5.5(b) is
followed by the top view of pore system and top and bottom layers of the pore system.
The surface of the disks is quite flat and very thin. Both blocks of polymer system
segregated well so that both blocks are clearly visible. Both blocks of the polymer
system segregated in the form of plate layers where one block is on top of the layer
whereas second is on bottom of layer of the disk. Despite of squeezes the domain size
both disks are not connected to each other and there is very narrow space between the
disks. However, in the system of equal size and length (d = 2.5 and h = 2.5) three
parallel flat disks appeared in the pore system perpendicular to the pore system as
shown in Figure 5.5(c) along with the interior-layers. It shows quite good agreement
with the result of parallel confinement, in which there were three cylindrical parallel
sheets of cylinders, the pore system shown in Figure 5.4(c). This shows that in
cylindrical confinements parallel confinement induces parallel cylindrical sheets of
lamellae and perpendicular confinement induces perpendicular lamellae sheets.
The last, two dimensional confinements’ results were investigated in cylindrical pore
geometry. In two dimensional confinements one of the monomer of polymer system has
preferential affinity to the pore surface and the pore system was confined between
parallel walls and perpendicular walls around the pore system. For two-dimensional
confinements here boundary conditions are applied in the radial direction and height z
direction. The interaction between walls and one of the blocks of the polymer system
was applied 2.0 in the pore system.
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Figure 5.6: Two dimensional confinement of cylindrical system with the pore sizes and the pore lengths
are (a) d = 1.0 and h = 1 (b) d = 1.0 and h = 1.6.
The simulation results were obtained under two-dimensional confinements show perfect
circular rings. The pore system shown in Figure 5.6(a) under two-dimensional
confinements a perfect circular cylindrical ring parallel to the pore system appeared.
The pore system with the increased size of pore also shows circular ring but here in this
pore system circular ring consists of two cylinder one on other. This shows that, under
the effect of two-dimensional confinements and the narrow size cylindrical pore,
lamellae segregated into cylindrical system in the pore geometry.
5.2.2. Cylindrical forming system confined in cylindrical pores
In this section, the diblock copolymer cylindrical forming system was investigated with
various cylindrical dimensions of pore system, pore radii and pore length by using the
CDS method functional in cylindrical coordinates. The polymer system was
investigated without confining surfaces (neutral surfaces) as well as preferential
attractive circular walls having affinity to the majority segment of the polymer system.
The results were also obtained by one-dimensional confinement and two-dimensional
confinements.
I. Cylindrical forming system confined in neutral surfaces
In this section, the results of diblock copolymer cylindrical forming system confined in
cylindrical pores are presented and discussed. The CDS parameters were used in the
computation are = 0.30, f = 0.40, u = 0.50, v = 1.5, B = 0.02, D = 0.50 A = 1.5.
Simulation results were obtained in various pore sizes by keeping the interior radius ar
fixed and the pore size of system was expanded by the increasing the exterior radius br
(a) (b)
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of the pore system. The length of the pore system was fixed at h = 4 and angular
coordinate varied 20 in the pore system. The pore size of the system is defined
by d = br - ar , where ar denote the interior radius of pore system and br denote the
exterior radius of the pore system. All the simulation results were carried out without
confining walls (neutral pores) around the pore system and were carried out on one
million time steps.
In the first case, the simulation results were obtained for cylindrical forming system in a
pore system in which the interior radius was fixed at ar = 3 and the exterior radius br of
the pore system were increased to expand the pore size of the system.
Figure 5.7: Cylindrical forming system confined in neutral surfaces, in a pore system with the interior
radius fixed at ar = 3, the pore sizes of the system are (a) d = 1 (b) d = 1.5 (c) d = 3.
The simulation results for cylindrical forming system in the pore system are oriented
parallel to the pore system in a small sized system. However, with increasing size of the
pore system the cylinders change their orientation perpendicular to the pore system. The
pore system d = 1 show cylinders parallel to the pore system in the pore geometry as
shown in Figure 5.7(a). While, increasing the size of the pore system d = 1.5 cylinders
became perpendicular to the pore geometry as shown in Figure 5.7(b). The bigger size
pore system d = 3 show most of the cylinder are perpendicular to the pore system as
shown by Figure 5.7(c). This shows that cylinder under the influence of the curvature
turn to be parallel to the pore system in smaller sized systems, while, in larger sizes
system due to nominal curvature they reorient themselves perpendicular to the pore
system. Furthermore, in the larger sized systems cylinders are still observed parallel to
(a) (b) (c)
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the pore system due the effect of the curvature. Cylindrical forming system in the pore
geometry also shows a few defects in the form of spheres. Cylinders in the pore system
show rhombus packing arrangement and Y-shape formation under the influence of the
curvature.
In the second case, the interior radius of pore was fixed at ar =5 in all simulation results,
while, the pore size of the system was increased by using various values of the exterior
radius br of the pore system. The cylindrical morphology was obtained without
interacting wall (neutral pores) and shows short straight cylinders and curved cylinders,
most of cylinders are perpendicular to the pore axis in the pore geometry. Short straight
cylinders which are perpendicular to the pore system form in the pore geometry due to
the minimal curvature effect. In the Figure 5.8(a), cylindrical system obtained in narrow
cylindrical channel (pore size d = 0.5), where short straight cylinders can be observed
oriented perpendicular the cylindrical axis. Short straight cylinders are hexagonally
packed as shown in zoom snapshot below the Figure 5.8(a). Increasing the pore size of
the system (d = 1) and decreasing length (h = 2.8) of the pore system we get similar
cylindrical system, however some short cylinders showed mixed orientation as shown in
Figure 5.8(b). In this system very few defects in the form of spheres were also observed.
The system shows parallel, perpendicular and oblique cylinders. For the pore system
obtained for dimensions of the pore (d = 1.5 and pore length h=4) shown in Figure
5.8(c) results shows the mobbed of short cylinders mostly oriented perpendicular to the
pore axis. This system also shows very few spheres and few cylinders with mixed
orientation. In this system a very few long cylinders observed with oblique orientation.
Micro domains vary in size, shape and orientation in the pore system.
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Figure 5.8: Cylindrical morphology in the neutral cylindrical pores with a pore system whose the interior
radius was fixed at 5ar and the pore size of systems are (a) d = 0.5 (b) d = 1.0 (c) d = 1.5.
I. Cylindrical system with interfacial surfaces
The value of at the boundary describes the preference of one copolymer block to the
surface of the pore system. For the interaction between wall and copolymer, we set the
strength of interaction between surface wall and one of polymer block at = 0.2. The
simulation results are obtained in different pore systems using one dimensional
confinement of parallel circular walls around the pore system.
In the first case simulation results were obtained in a pore system whose interior radius
was fixed at ar = 3 and the exterior radius br of the pore system was increased different
values to expand the pore size of the system. Surface interaction strength to the majority
segment of the polymers system was applied with = 0.2 in the pore system.
(a) (b) (c)
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Figure 5.9: Cylinder system under interfacial surfaces in a pore system with the interior radius fixed at
3ar and the pore sizes of system are (a) d = 1 (b) d = 1.5 (c) d = 2 (d) d = 2.5 (e) d = 3.
The cylinder forming systems under influence of surface interaction in a pore system
which has the interior radius fixed at ar = 3 show cylinders are wrapped around the pore
system and perforated morphology in the pore system. In the narrow channel size pore
system with pore size d = 1 pore system displays single layer of polymer system with
perforated morphology as shown in Figure 5.9(a). The pore size d = 1.5, system show
two-layered system where straight cylinders wrapped around the pore system as shown
in Figure 5.9(b) with interior layer of the system. This pore system also displays few
defects in the form of perforated holes, short cylinders and spheres in the pore
geometry. The pore system with pore size d = 2, show three layered system where
straight cylinders are wrapped around in the pore system and having similar defects of
perforated holes and spheres in the pore geometry as shown in Figure 5.9(c) along with
interior layers of the system. Wrapped cylinders forms due to the curvature effect in the
pore system. Similarly the pore system d =2.5, show four layers of the micro domains
(a) (b)
(c)
(d) (e)
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in which cylinders are also wrapped around the pore system and this system is less
defective as compare to other systems, pore system is shown in Figure 5.9(d). As we
increased the size of the pore system to d = 3, we observed cylinders with mixed
orientation including parallel to the pore system and more defective system as shown in
Figure 5.9(e). Simulation results show that due to the interplay of both curvature and
confinement cylinders are wrapped around the pore system in the pore geometry.
In the second case, the pore system was designed by keeping the interior radius fixed at
ar = 5 and pore size was increased by taking different valued of the exterior radius br of
the pore system. The simulation results were obtained for characteristic case in which
the pore wall is attractive to the majority of the block of the polymer system in the pore
geometry.
Figure 5.10: Cylindrical forming system under surface interacting parameter 2 , the interior radius of
pore system is fixed at 5ar , the length of pore was fixed at h = 6, 20 and the pore sizes are (a) d
= 0.9 (b) d= 1.0 (c) d = 1.5 with the interior layer (d) d = 2 (e) d = 2.5 (f) d = 3.
(a) (b)
(c)
(d)
(f)
(e)
87
The results under geometrical confinement show that under the influence of preferential
attractive circular walls cylinders show mixed behaviour. The curved cylinders form
different formations under confinement U-shape cylinders, Y-shape cylinders and W-
shape cylinders. Perforated lamellae PL morphology observed here in the system for the
case when pore size was d = 1 under geometrical confinement.
In the narrow cylindrical pore size system (d = 0.9), we get the system in single-layer
and micro domains became compressed and loosed their structural formation only very
few micro domains able survive in their structural morphology in the pore system as
shown in Figure 5.10(a). With pore size (pore size d = 1.0) system turned into
perforated morphology with few cylinders oriented parallel to the pore system and
oblique Figure 5.10(b) shows the results. In this case also we get a single layered
system. Increasing the pore size of the system by expanding the exterior radius of pore
(d = 1.5), we get bi-layered system inside the pore system. In this system a very few
perforated holes are observed and rest of the system displays curved cylinders.
Microdomains are oriented parallel to the pore axis and oblique. Figure 5.10(c) shows
the system with interior layer adjacent. The microdomains were observed are long
cylinders with few defects of spheres in the pore geometry. There are also some cross
sectional cylinders apparent in interior layer of the system. Increasing further size of the
system (d = 2.0) shown in Figure 5.10(d) system is more defective as compared to
previous one in the proe system. The cylinders in the pore system are diversely oriented.
For the pore size (d = 2.5) system shows three layers of cylinder system with mixed
orientation in the pore system as shown in Figure 5.10(e) along with two interior layers.
In this system orientation of curved cylinders remained diverse this mean that
preferential attractive wall effect is weak in the pore system. Increasing the pore size bit
more (d = 3) system show cylinders oriented perpendicular to the pore system in the
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pore geometry as shown in Figure 5.10(f). This shows that long sized pore system
curvature and confinement influence is nominal.
The cylindrical forming system investigated in narrow channels keeping the fixed pore
size d = 1 by keeping the interior radius of pore fixed at ar =5, the exterior radius of
pore at fixed at br =6, while angular coordinate varied 20 however, length of
pore varied from h=1 to different values ranging from h=1 to h=4.0 are shown in
Figure 5.11. Simulation results are obtained on one million time steps with preferential
attractive walls. Length of cylindrical pore was varied to check influence of
confinement on different layers of cylinders. The circular wall interaction with one of
block was set at 2.0 in the pore system. The cylindrical system was investigated by
one dimensional confinement and two dimensional confinements. In one dimensional
confinement circular walls were placed parallel to the pore axis and two dimensional
confinements in addition to the parallel to the pore walls the top and bottom covering
walls were placed confining the pore surface perpendicularly through cross sections of
the pore system. Figure 5.11 shows the results obtained with various lengths of pore.
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Figure 5.11: Cylindrical forming system with fixed pore size d =5, whereas, the height was varied for
different values of lattice.
In the first case when was h=1 simulation results under one dimensional confinement
show most of the cylinders are oriented parallel to the cylinder axis due to the wall
interaction. In one dimensionally confined system very few perforated holes can be
observed. Interestingly with two dimensional confinements we obtained just one
cylinder perfectly in the circular ring shape. Perfectly a circular ring cylinder obtained
due to the two factors one is curvature effect and other is due to the interacting circular
walls. Further height of pore increased h=1.6 to study bi-layered cylinder with one
dimensional cylinder oriented parallel to the axis of cylinder, parallel to the
circumference of cylinder and oblique. However with two dimensional confinements we
obtained the ring containing two circular cylinders one on other and between these
cylinders perforation morphology observed. For the h=2.2 system three layers of
cylinders not observed in two dimensional confinement which shows third cylinder in
between the two cylinder is distorted by internal energies and due to this perforated
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morphology is developed between the cylinders.. As we moved forwarded by increasing
height, we find more perforated morphology appeared in one dimensional confinement
and in two dimensional confinements perforated morphology is observed between
parallel to the pore cylinders. Perforated morphology was also observed in planer and
cylindrical confinement [10]. Similar patterns of parallel cylinders to the axis of pore
and perforated morphology under two dimensional confinements can be observed in
[87] where you can see there in Figure 3(b) and furthermore, confinement distorts
hexagonal packing of the cylinders. Similar nanostructure were obtained in nanotubes,
Chen et al [88] explained these nanostructures shown in Figure 2 therein, as uniform
wormlike nanostructures entangled each other to form perforated holes morphology.
The simulation results for the cylindrical forming system were also obtained by
increasing interaction strength 4.0 between the surface walls and majority of the
block of the polymer system in the pore geometry. One-dimensional confinement is
used in this system to investigate the effect of preferential attractive circular wall
parallel to the pore. Simulation results are carried out on one million time steps. The
simulation results for the cylinder system are obtained in a pore system with interior
radius fixed at 5ar and exterior radius br of the pore system varied to expand the pore
size of the system. The length of the pore system is also fixed at h = 4 however, in one
system it is changed which is mentioned in the figure below. Snapshots of the
simulation results are shown in figure 5.12.
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Figure 5.12: cylindrical system obtained with interaction parameter 4.0 , the pore size and the pore
lengths respectively are (a) d = 1.6 and h = 1.6 (b) d = 1.5 and h = 4 (c) d = 2 and h=4.
The first system obtained with pore size d = 1.6 and pore length h = 1.6 shown in
Figure 5.12(a) with zoomed snapshots adjacent to the image and the interior layer of the
system. System shows mostly perpendicular to the pore tilted cylinders in circular
alignment with very few parallel to the pore cylinders. Perpendicular to the pore
cylinders are due the influence of preferential attractive wall in the pore system. Parallel
to the pore cylinders are tilted cylinders emerged due to the strong circular wall
interaction 4.0 and in previous simulation results with week wall interaction we
observed curved cylinders in the pore system. These tilted cylinders appeared in both
layers of the pore system. For the pore system size d = 1.5 and pore length h = 4 system
there are curved cylinders appear in the pore system as shown in Figure 5.12(b).
Furthermore increasing the size of the pore system diversely oriented cylinders appear
in the pore system as shown in Figure 5.9(c). The experimental study [86] shows that in
the cylinder forming system under the preferential wetting of the pore walls
microdomains align along the pore axis in cylindrical confinements.
Interior
layer
(a)
(b) (c)
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5.2.3. Spherical forming system confined in cylindrical pores
In this section, a study was carried out on spherical forming system in cylindrical pores
using CDS method in cylindrical coordinates system. The Spherical polymer system
was investigated using various cylindrical pore dimensions without confinements
(neutral surfaces) and with preferential affinity to the surface of the pore system of one
of the monomer of the polymer system. All the simulations were carried out on one
million time steps.
I. Sphere forming system confined in neutral pores
First of all, the sphere forming system was obtained without confinement of the pore
walls in the pore system. The sphere morphology of diblock copolymer system were
obtained in a hollow cylindrical pore system. In the first case simulation results are
obtained by using following CDS parameter system with low temperature parameter in
the pore system.
Table 6: CDS parameter system with low temperature for sphere forming system
f u v B D A
0.20 0.40 0.38 2.3 0.01 0.50 1.50
In the first case for neutral pore system, the interior radius of the pore system was fixed
at ar = 3 whereas the exterior radius br were increased by five grid points to expand the
pore system. In the pore system d describes the pore size of the system which is the
difference of the interior and the exterior radius of the pore system.
93
Figure 5.13: Sphere forming system in neutral cylindrical pore systems with low temperature CDS
parameter system, the interior radius of pore system was fixed at ar = 3 and the pore sizes are (a) d =2.5
(b) d = 3.0.
The results in the pore system show a lot of defects arising in the form of short cylinders
in the pore system as shown in Figure 5.13. There are spheres which vary in size
throughout the pore system. This may be due to the influence of the curvature because
in the outer layer system is less destabilized as compare to the interior layer where
system is almost destabilized in the pore system.
In the second case, simulation results were obtained using the CDS parameter system
with low temperature for sphere system in the pore system whose interior radius is fixed
at ar =5 and exterior radius br varied to expand the pore system. Similarly d = ab rr
defines the pore size of the pore system.
Figure 5.14: Sphere system obtained by low temperature CDS parameter system, the pore system has the
interior radius fixed at ar =5 and pore sizes in the images are (a) d = 0.5 (b) d = 1 (c) d = 1.5.
The sphere forming system obtained in the pore system with the interior radius fixed at
ar =5 also show defects in the form of short cylinders in the pore system as shown in
Figure 5.14. Two spheres merge to gather to form short cylinders in the pore system.
Short cylinders formed by the merging two spheres are look like double dumbbell in the
pore system and the long cylinders are formed by the merging of three or more than
(b) (a)
(a) (b) (c)
94
three spheres in the pore geometry. There are also observed defects in the form of long
curved cylinders and straight cylinders in the pore geometry. However, system show
better stabilized sphere system in the pore geometry as compared to the pore system
whose interior radius was fixed at ar = 3. This further strengthening the argument that
defects arising in the sphere system with low temperature was due to the curvature
influence in the pore system. The spheres which maintain their structural morphology
do not vary too much in shape and size as compared to previous pore system with
interior radius fixed at ar =3. We can observe some perfect sphere system in the pore
geometry.
I. Sphere system with modified CDS parameter system in cylindrical pores
The sphere forming system is also achieved in the pore geometry by using modified
CDS parameter system with higher temperature. The modified CDS parameter system
for sphere forming system is achieved by extensive simulation work, while searching
for modification of cylindrical forming system. The search started by taking different
variations around the temperature parameter 30.0 which was fixed for the cylinder
system. The temperature parameter is chosen for modification because mainly the
temperature parameter and volume fraction f play important role in the morphology
formation of diblock copolymers systems. The volume fraction f = 40 is fixed for both
cylinder system and sphere system. The search domain was 530.0 with fixed
change in each time given to 30.0 be the fixed number ±0.1, so there are 100
simulations were carried out in the search of modification of CDS of parameters system.
Unfortunately, we couldn’t find any best CDS parameter system for cylindrical system
but we did find a CDS parameter system for sphere forming system. The modified CDS
parameter system for sphere system is shown in table 7. Note that values for CDS
parameters u, v and B were the same as used for cylinders system because we were
95
looking for modification of cylinder system and using CDS parameters for cylindrical
system in the mean time we arrived at the sphere morphology on 25.0 .
Table 7: Modified CDS parameter system for sphere forming system
f u v B D A
0.25 0.40 0.50 1.5 0.02 0.50 1.50
In the first case, simulations results are obtained with neutral surface in a pore system
with the interior radius fixed at ar =3 while, the exterior radius br of the pore system
increased by 0.5 grid points to expand the pore size of the system.
Figure 5.15: Sphere system obtained by modified CDS parameters system with high temperature in a pore
system with the interior radius fixed at ar = 3 and the pore sizes are (a) d = 1 (b) d = 2 (c) d = 2.5.
The simulation results were obtained by modified CDS parameters for sphere systems in
a pore system with interior radius fixed at ar = 3 show perfect sphere system packed in
cylindrical pores as shown in Figure 5.15. There is no any defect in the form of short
cylinders and all micro domains are perfect spheres having similar size and shape in the
pore system. There are various packing arrangement of spheres observed in the pore
system like rhombus, pentagon and quadrilateral. Simulation results are very well
ordered and stabilized as compare to simulation results obtained with low temperature
CDS parameter system for sphere forming system shown in previous section. This
shows that increasing temperature parameter and B which defines the chain length
dependence of free energy functional played a role in stabilizing the sphere forming
(a) (b) (c)
96
system. This modification of CDS parameter system for sphere forming system will
further pave the way for predicting new morphologies in the future.
In the second case, sphere system obtained with modified parameter system in apore
system having interior radius fixed at ar = 5.0 and exterior radius br of the pore system
varied to expand the size of the system, therefore pore size of the system becomes d =
br - ar . The length of the pore system represented here by the h = 4 which is fixed as
well in the pore system, whereas angular coordinate of the pore system varied
20 throughout the pore system.
Figure 5.16: Sphere system obtained with modified CDS parameter system in a pore system whose
interior radius were fixed at ar = 0.5and the pore sizes of the systems are (a) d = 0.5 (b) d = 1 (c) d = 1.5
(d) d = 2 (e) d = 2.5 (f) d = 3.
The simulation results for sphere forming systems obtained by modified CDS parameter
system in a pore system with interior radius fixed at ar = 5.0 also show perfect spheres
packed into cylindrical pore system without any single defects of the short cylinders as
shown in Figure 5.16. The pore system with size d = 0.5 show spheres are perfectly
packed in the diagonal strips and anti-diagonal strips in the pore surface as shown in
Figure 5.16(a) by dashed lines. Sphere system due to the packing along diagonal and
anti-diagonal packing also show square packing arrangements in the pore geometry. On
(a) (b) (c) (d)
(e) (f)
97
the other hand spheres packing can be seen as along the vertices of the concentric
squares in the cylindrical pore surface. The spheres in the outer layer (cut off in half) are
shifted slightly with respect to the spheres in the interior layer of the nanostructure.
However, by increasing the pore size of the system this diagonal packing of spheres
disappeared in the pore system due to the size effect. This shows that except the narrow
channel size system d = 0.5, curvature does not influence on the spheres system in the
pore system as remaining system shown in Figure 5.16 did not show any significant
change in the pore system. The sphere system under modified CDS parameter system
achieves its minimum energy rapidly as compared to the low temperature CDS
parameter system for sphere forming system. The sphere system in cylindrical pores
shows micro domains are jam-packed as shown in Figure 5.16(b-f). This shows that,
there is no size effect and curvature effect on the system. The system shows no
alignment of the micro domains in the cylindrical pore system. All the spheres are
packed in single layered cylindrical pore system. This system of modified parameters
also works very well for CDS system in Cartesian coordinate system.
II. Sphere system with interfacial surfaces
The sphere forming system was obtained in cylindrical pore system by applying
preferential affinity to the surface for one of the monomer of the polymer system. The
modified CDS parameter system for the sphere forming system are not working under
geometric confinements because under geometric confinement micro domains in the
system becomes squeezed down, lose their structure and get close to each other. This
may be due to the high temperature and interaction of surface. Therefore for
confinement case we inserted the Previous CDS parameter system with low temperature
shown in table 8.
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Table 8: CDS parameter system for sphere forming system for low temperature
f u v B D A
0.20 0.40 0.50 2.30 0.01 0.50 1.50
In the first case, preferential attractive circular walls parallel to the pore system applied
in a pore system with the interior radius fixed at ar = 3 and the exterior radius ar of the
pore system increased by 5 grid points to expand the pore system. The interaction
strength set was at 2.0 between the pore walls and the majority segment of the
polymer system in the pore geometry.
Figure 5.17: Sphere system with interacting parallel circular walls having preferential affinity with
majority segment of the polymer system, in a pore system with interior radius of the pore were fixed at ar
= 3 and the pore sizes of the system are (a) d = 1.5 (b) d = 2 (c) d = 2.5.
The sphere system confined by circular parallel walls in a pore system with interior
radius fixed at ar = 3 show sphere packed in the pore system with defects in the form of
cylinders formed by merger of two or more spheres in the pore geometry as shown in
Figure 5.17. The pore system with pore size d = 1.5 show single layer packed with
sphere along with defective cylinder in the pore system as shown in Figure 5.17(a).
However, the pore system having size d = 2 show two layers of pore system packed
with spheres in the pore geometry as shown by the Figure 5.17(b) along with interior
layer of the system. This pore system shows that interior layer of the pore is more
defective as compared to outer layer of the pore system which shows that interior layer
of the system is more defective due the influence of the curvature on the sphere forming
system. The pore system with pore size d = 2.5 also shows defective sphere system as
(a) (b)
(c)
99
shown in Figure 5.17(c). This pore system is a less defective compared to the previous
pore size system which shows that increasing the size of the system we get a less
defective system.
In the second case, simulation results were obtained with preferential attractive walls
parallel to the pore system having affinity to the majority of the block of the polymer
system in the pore geometry. The interior radius of the pore system kept fixed at ar = 5
and the exterior radius br of the pore system varied to expand the pore size of the
system. The interaction strength between the circular walls and majority segment of the
polymer system applied 2.0 in the pore system.
Figure 5.18: Sphere system under interfacial cylindrical surfaces and size of pore systems are (a) d = 1.5
(b) d = 2 (c) d = 2.5 (d) d = 0.3.
Under geometric confinements, micro domains show circular alignment around the
centre of the pore. However, the structure shows some defects in the form of short
cylinders formed by merger of two spheres. For the pore size d = 1.5, pore system
shows single layered system and micro domains are packed in the pore as shown in
Figure 5.18(a). The pore system with pore size d = 2, shown in Figure 5.18(b) display
spheres packed in concentric circular rings parallel to the pore system. In this two
(b)
(c) (d)
(a)
100
layered system we observed very few defects in the form of short cylinders. The pore
system with size d = 2.5, shows micro domains are packed in three layers of concentric
circles parallel to the pore system. This system also shows very few defects in the form
short cylinders. This shows that under geometrical confinement sphere forming systems
tend to form circular alignments in concentric circles parallel to pore system. Results are
also obtained by applying interaction strength 4.0 between the majority segment of
the polymers system and cylindrical walls in the pore system but we observed no
significant change in the results.
For the second case, interacting walls were applied perpendicular to the pore through
cross section of cylindrical pore. In this system polymer is confined in cylindrical pore
and interacting walls covering the top and bottom of the pore system. Strength of
interaction between walls and one of the polymer blocks set at 2.0 .
101
Figure 5.19: Sphere system under geometric confinement applied through cross sections of cylindrical
pore system with the pore size and the pore length are respectively (a) d = 1.5, h = 1.2 (b) d = 2.5, h =
2.5 (c) d = 3.0, h = 3.0.
Under perpendicular confinement applied through the top and bottom cross sections of
cylindrical pore system micro domains show perpendicular packing to the pore system.
For the system with pore size and length respectively d = 1.5, h = 1.5, display spheres
packed in a single layer perpendicular to the pore system as shown in Figure 5.19(a)
along with top view of the image. There are some defects as well in the form of short
cylinders. Increasing size and length respectively d = 2.5, h = 2.5 of the system, pore
system shows spheres packed in bi-layered system perpendicular to the pore system as
shown in Figure 5.19(b). For this size and length of the pore system some defects are
(a)
(b)
(c)
102
observed in the pore system. Similarly, for the system size and length respectively d =
3.0, h = 3.0, spheres are packed in three layers perpendicular to the pore system, pore
system shown in Figure 5.19(c). This shows that geometric confinement perpendicular
to the pore system induces the packing of micro domains in layers perpendicular to the
pore system. Increasing the strength of interaction between walls and polymer blocks
we get more defective system. The simulation results were also obtained with the very
strong interaction parameter was set at 4.0 , we get the pore system with very few
more defects in the system not shown here.
The simulation results for sphere forming system were also obtained with two
dimensional confinements applied on the cylindrical pore system. Two dimensional
confinement was applied by the parallel walls and perpendicular walls around the pore
system. Parallel walls are imposed on the cylindrical pore system by applying the
boundary conditions on the radial coordinate of the cylindrical coordinate system.
Perpendicular walls are imposed on the pore system by the applying the boundary
conditions on the z coordinate system of cylindrical coordinate system. The pore system
is confined between parallel circular walls and pore system is capped by top and bottom
interacting walls. The strength of interaction between surface and one of the blocks of
polymer system used in simulation was set at 2.0 .
103
Figure 5.20: Sphere system under two dimensional confinements with system size and length are
respectively (a) d = h = 1.5 (b) d = h = 2.5 (c) d = h = 3.0.
The results obtained under two dimensional geometric confinements shows that spheres
are induced in circular alignments in the cylindrical pore system. In the first case, two
dimensional confinements applied on the narrow system with size and length
respectively d = h = 1.5 as shown in Figure 5.20(a). In this system due to the confining
effect some spheres merged to gather and formed cylinders. There are seven cylinders
formed by merging two or more spheres. There are two short cylinders formed by
merging two spheres, however there are five medium cylinders formed by the merging
of three or more spheres. The remaining survived spheres are in a perfect circular
alignment with cylinders around the centre of the pore system. However, the surviving
spheres are interconnected to each other in the pore system. The story of the pore
(a)
(b)
(c)
104
cylindrical pore system under two dimensional confinements is much different than the
narrow sized pore system. The pore system with size and length respectively d = h =
2.5, show that the spheres are in perfect circular alignment in concentric circles and
circles one upon each other as shown in Figure 5.20(b). The microdomains are induced
in pore system by four circular layers parallel to the pore system as well as
perpendicular to the pore system due to the two dimensional confinements. The
cylindrical pore system of size and length d = h = 3.0 under two dimensional
confinements also shows spheres are induced in the cylindrical pore system in circular
rings parallel to the pore system as well as perpendicular to the pore system as shown in
Figure 5.20(c). This system size also shows very few defects under two dimensional
confinements as compared to system with same size under one dimensional
confinement. There are in total nine circular rings induced spheres in the cylindrical
pore system. However, there are also very few defects of short cylinders formed due to
the merger of two spheres. Under two dimensional confinements we obtained less
defected and more ordered system in the cylindrical pore system due to the interplay of
surface interaction and curvature effect. Two dimensional confinements are working
well in the cylindrical pores for the sphere forming system as compared to cylindrical
and lamella.
5.3. Summary
Lamella foming system confined in the neutral cylindrical pore show mixed orientations
like helical, strip lamella and lamella sheets perpendicular to the pore surface. However,
in the presence of the interacting surfaces concentric lamella were induced in the pore
geometry. Results show that under the influence of the both curvature and interfacial
surfaces lamella tends to form concentric layers. Cylinder forming system under
geometric confinements of the neutral cylindrical pore show mixed orientations of
105
microdomains, perpendicular to pore surface and parallel to the pore surface. However,
in the presence of interfacial cylindrical surfaces microdomains show parallel to pore
surface orientation and wrapped around the pore system. In the narrow pore size system
perforated morphology and parallel to the pore cylinders were observed in the pore
geometry. In the short length pore surface we obtained standing cylinders which were
packed in the concentric circles in the pore system. The sphere forming systems were
obtained with a new set of CDS parameters with high temperature. In this case we
obtained a perfect sphere system in the cylindrical pores without any defective
cylinders. Sphere forming system confined in the cylindrical pore with interfacial
surfaces show concentric circles packing arrangements.
106
CHAPTER SIX
6. Diblock copolymer system confined in spherical pores
6.1. Introduction
This study investigates the nonstructural formation of block copolymers on the surface
of a sphere in the spherical coordinate system. In this study, investigations were carried
out for classical morphologies of diblock copolymer systems lamella, cylindrical and
spherical forming systems in the hemispherical pore shell. Lamella, cylindrical and
spherical morphologies are investigated under geometric confinement of spherical pores
with neutral surfaces and with interacting surface walls having affinity with the majority
segment of the polymer system. In the case of interfacial surfaces, symmetric boundary
conditions (each interface has same preference for the selected segment of the polymer
system) were applied on the pore interfaces. The results were obtained in the pore
surfaces with the interior radius ar fixed and the exterior radius br was varied to expand
the pore size of the surfaces. In the pore shell, the simulations were carried out by radial
step 1.0r , with polar angular step 360/2 , with azimuthal angular step
180/ . The CDS simulations are carried out with time step 1.0t . In order to
obtain results in the hemispherical shell by avoiding singularities, the radial coordinates
are restricted by ba rrr , similarly angular and azimuthal coordinates are also
restricted by ,0 and 0 respectively. The pore size of the surface is defined
by the relation ab rrd , which is the difference between the exterior radius of the pore
system and the interior radius of the pore system. All simulations are carried out for one
million time steps. All the CDS simulations were carried out by initializing the order
parameter Ψ in the range of values ±1.
107
6.2. Results and discussion
Using CDS simulations employed in spherical coordinate system, investigations were
carried out for lamella, cylinder and sphere forming block copolymers with various pore
radii of hemispherical shells. The confinement induced and curvature influenced
nanostructures were obtained with neutral spherical surfaces as well as with interfacial
spherical surfaces. In the interfacial results, the majority segment of the polymer system
is chosen to have affinity with the pore walls. In order to investigate the curvature
influence, results were obtained with various pore thicknesses of the pore surfaces in the
pore geometry.
6.2.1. Lamella forming system confined in spherical pores
In this section, lamella forming system is investigated in the spherical pores. The results
were obtained with the different pore sizes of the surface by keeping the interior radius
ar fixed and by varying the exterior radius br to expand the pore surface are shown and
discussed.
I. Lamella forming system with neutral surfaces of the spherical pores
The results were obtained and discussed for asymmetric lamella forming system with
neutral surfaces in the spherical pores using the CDS parameter system and these
parameters are shown in table (9).
Table 9: CDS parameter system for asymmetric lamella forming system
f u v B D A
0.36 0.48 0.38 2.30 0.02 0.70 1.50
In the first part, the simulation results were obtained in a pore surface with the interior
radius were fixed at 3ar and the exterior radius br of the pore surface varied to expand
the pore size of the system.
108
Figure 6.1: Asymmetric lamellae system in spherical pores with neutral surfaces, the interior radius of the
pores were fixed at 3ar and the pore sizes of the systems are (a) d = 0.5 (b) d = 1 (c) d = 1.5 (d) d = 3.
Asymmetric lamella confined in spherical shells having the interior radius fixed at
3ar shows concentric lamellae sheets with perforated holes. In the sphere shell size d
= 0.5, there is single a parallel lamellar sheet with perforated holes as shown in Figure
6.1(a). The pore surface d = 1 displays onion-like structure of lamella with standing
lamella packed in the interior sheet of the structure as shown in Figure 6.1(b). There are
three parallel concentric lamellar sheets in the pore surface. As we increasing thickness
of the pore surface d = 1.5, sphere shell induces four concentric lamellar sheets with
few standing lamella packed in the interior sheet of the pore surface as shown by the
Figure 6.1(c). As the thickness of the pore surface is increased, concentric lamellae
sheets tend to deform into a stacked lamella system. The pore system having the size of
pore d = 3, shows mixed concentric lamellar sheets and stacked lamella as shown in
Figure 6.1(d). The pore system displays seven layers of microdomains. However,
stacked lamella were observed only on the outer layer of the pore surface. This shows
that by increasing the size of the spherical shell concentric lamella switch to the stacked
lamella morphology. Recently, onions-like structures were also obtained by
(a) (b)
(c) (d)
109
experimental study [89]. Similar Onion-like structures had also been observed by the
experimental work conducted by transmission electron microscopy [31] as shown in
Figure 7 there in and [68] shown in Figure 1(c) there in. However, this onion-like
morphology was achieved by the confinement of the pore surface but here we obtained
the results with neutral surfaces. There are also similar, parallel to the pore lamella in
the planar thin films were obtained in the experimental work [90]. These onions-like
structures were also reported by the simulated annealing Monte Carlo simulations [91]
by applying strong surface preference Figure 1(a) there in. The study was also
conducted by using the CDS computational method, where block copolymers were
grown in planar surfaces and shown on spherical pores [8]; they also observed onion-
like nanostructures in the spherical pore shell with one-dimensional confinement. The
onion-like concentric alternating layered lamella morphology was obtained by preparing
nanoparticles confined in the spherical surfaces from PS-b-PI diblock copolymer [85],
where results are shown by Figure 1 and Figure 3(b) are clearly consistent with the
results obtained in this study as shown in Figure 6.1.
In the second case, results for lamella system are obtained and discussed by keeping the
interior radius of the pore surface fixed at 5ar and the thickness of the pore surface
expanded by the increasing the exterior radius br of the hemispherical shell.
110
Figure 6.2: Asymmetric lamella system in the pore surface with the interior radius fixed at 5ar and
shell sizes are (a) d = 0.5 (b) d = 1 (c) d = 1.5 (d) d = 3.
The asymmetric lamella confined in the spherical shell with the interior radius of the
pore surface fixed at 5ar shows nanostructure evolution of the lamella system from
the concentric lamella to the stacked lamella morphologies with respect to the pore size
of the system. The spherical pore shell with a thickness of the pore surface of d = 0.5,
displays two concentric lamellae sheets with the smooth surface as shown in Figure
6.2(a). However, the spherical shell with pore size d =1 shows a mixed structure of
standing lamella, stacked lamellae and a few clusters of lamellar sheets as shown in
Figure 6.2(b). The pore surface d = 1.5, shows fully evolved stacked lamella
morphology as shown in Figure 6.2(c). Increasing the thickness of the pore system d =
3, it can be observed stacked morphology of the lamella system as shown in Figure
6.2(d). Results show that under the influence of the curvature, the lamella system forms
concentric parallel sheets and under the weak curvature effect lamella evolves into the
stacked morphology. These stacked lamellar systems obtained here, are consistent with
the stacked lamellar system which were obtained by the experimental work [68] shown
in Figure 1(a) there in, whereas, concentric lamella or onion-likes structures are also
(a) (b)
(c) (d)
111
consistent see Figure 1(c) there in. The stacked lamellar system were also reported by
the computational work [91] as in Figure 1(a) there in.
In the following, results were obtained with the interior radius of the spherical pore
surface was fixed at 7ar and the size of the spherical shells extended by the exterior
radius br of the pore surface geometry.
Figure 6.3: Asymmetric lamella system in spherical pore with the interior radius of the pore system fixed
at 7ar and the spherical pore sizes are (a) d = 0.5 (b) d = 1 (c) d = 1.5 (d) d = 3.
The lamella system in the spherical surface with the interior radius fixed at 7ar shows
formation of the nanostructure from mixed morphology to stacked morphology. The
narrow pore surface d =0.5 shows a so called standing-up lamella system with mixed
curved patterns in the pore surface as shown in Figure 6.3(a). By increasing the surface
size d =1, the pore surface displays stacked lamella morphology on the top and bottom
of the pore surface, while in the middle of the pore surface, there are standing-up
lamella with parallel and perpendicular orientation as shown in Figure 6.3(b). The pore
systems shown in Figure 6.3(c,d) show stacked lamella morphology developed under
weaker curvature influence in the pore system. Results show that lamella in the
(a) (b)
(c) (d)
112
spherical pore geometry evolves from mixed morphology to the stacked morphology in
the pore surface system with respect to the size.
II. Asymmetric lamella system with interfacial surfaces
In this section, we consider the asymmetric lamella forming system confined by similar
interfaces with preferential affinity to the majority segment of the polymer system in the
spherical pore surface. The interaction strength between the pore interfaces and majority
block of the polymer system which was applied is 2.0 .
In the first case, we consider pore surface with the interior radius fixed at 3ar and the
pore surface extended by the exterior radius br of the pore system.
Figure 6.4: Lamella systems with interfacial surfaces, the pore surface having the interior radius fixed at
3ar and the pore sizes are (a) d = 1 (b) d = 1.5.
The lamella system confined by the similar interfaces having the affinity to the majority
segment of the polymer system in a pore surface with interior radius fixed at 3ar
shows concentric lamella parallel to the pore surface. In the pore surface with pore sizes
d = 1, shows mixed lamella system where standing-up lamella is packed inside the
womb of the outer layer of the concentric lamella sheet and a few spots of the perforated
morphology on the interior layer of the lamella sheet in the pore surface as shown in
Figure 6.4(a). This type of the mixed lamella is a novel nanostructure where standing
lamella and concentric lamella coexist to gather in the spherical pore geometry.
However, the pore surface with the pore size d = 1.5 shows concentric lamella sheets as
shown in Figure 6.4(b). There are three lamella sheets in the pore surface with a smooth
(a) (b)
113
exterior sheet and there are still very few spots of the perforated morphology on the
interior lamellar sheet in the pore surface. Results show that the lamella under the
influence of the concentric surface confinement in the spherical pore surface induces the
structure in the form of the lamellar sheets which are parallel to the pore surface.
In the second case, we consider the pore surface with interior radius fixed at 5ar and
the exterior radius of the pore surface varied to expand the pore size of the system. The
polymer system is confined by the parallel spherical surface having affinity to the major
block of the polymer system in the spherical shell.
Figure 6.5: Lamella systems confined by interfacial interfaces, with pore surface having the interior
radius fixed at 5ar and the pore sizes are (a) d = 1 (b) d = 1.5.
The results for the asymmetric lamella forming system confined by the symmetric
parallel spherical interfaces having preferential affinity to the majority block of the
polymer system show concentric lamellae sheets parallel to the pore surface in a pore
surface with the interior radius of the pore fixed at 5ar . The pore surface with the pore
size d = 1, induces a single smooth surfaced lamella sheet parallel to the pore surface in
the pore geometry as shown in Figure 6.5(a). The spherical pore surface having the pore
size d =1.5, displays two concentric parallel lamella smooth surface sheets in the pore
surface as shown in Figure 6.5(b). Results predict concentric smooth lamellae sheets
parallel to the pore surface in the pore geometry under confined parallel interfaces in the
pore geometry.
In the third case, results are obtained and discussed with the interior radius of the pore
shell fixed at 7ar and the exterior radius br of the pore surface varied for five grid
(a) (b)
114
points to expand the pore size of the surface. The interacting similar interfaces parallel
to the pore surface applied on the polymer system have affinity with the majority
segment of the polymer system.
Figure 6.6: lamella systems confined by interacting interfaces, with the pore surface having the interior
radius fixed at 7ar and the pore sizes are (a) d = 1 (b) d = 1.5 (c) d = 2.0 (d) d = 2.5 (e) d = 3.0.
The pore surface with the interior radius fixed at 7ar , the polymer system was
confined by parallel spherical interfaces in spherical pore geometry and shows evolution
of different morphologies of lamella forming system. In the first case, the pore surface
with pore size d = 1, shows interesting mixed nanostructures which induce lamellar
(a) (b)
(c)
(d)
(e)
115
sheets, so called standing-up lamellae, perpendicular cylinders, spheres and perforated
morphologies in the spherical pore surface as shown in Figure 6.6(a). Results predict
coexistence of the lamellar, cylindrical and spherical forming systems in the spherical
pore surface confined by the similar interfaces which is quite a novel nanostructure in
the field. The hybrid nanostructures were also reported confined by dissimilar interfaces
in planar thin films [92] which are a different case from spherical geometry. However,
hybrid morphology of the parallel and perpendicular lamella confined in spherical pores
under dissimilar interfaces, were reported in the computational work [8] but this work
shows only the coexistence of the parallel and perpendicular morphologies of lamella
system. The pore surface with the pore size d = 1.5, induces smooth lamellar sheet with
perforated holes on top and bottom of the surface as shown in Figure 6.6(b). However,
the pore surface having a pore size d = 2.0, induces two lamellar sheets with a few
perforated holes on the interior lamellar sheets as well as exterior lamellar sheets in the
spherical pore surface as shown in Figure 6.6(c) along with an interior layer. The
interior layer of the structure shows that the both layers are also interconnected by the
few neck-bridges. Perforated holes are results of the bilayer lamellar sheets. For the
pore surface with pore size d = 2.5, induces stack lamellar system on top and bottom of
the pore surface and perforated lamella in the middle of the layers in the pore surface as
shown in Figure 6.6(d) along with interior layer on the right hand side of the image.
Interior layer of the pore system shows that layers are interconnected by the neck-
bridges to each other in the pore geometry. The pore surface having pore size d = 3.0,
induces the mixed structure of lamellar sheets, a few perforated holes and stacked
lamellar on the top and bottom surfaces of the spherical pore system as shown in Figure
6.6(e). Results were also show that the lamella system under wetting surfaces evolves
into the stacked lamellar morphology from lamellar sheets morphology with respect to
the size of the pore surface. Results were obtained with various sizes of pore surface
116
show evolution of the lamellar morphologies from a hybrid system to lamellar sheets,
lamellar sheets to the perforated holes, perforated holes to lamellar sheets again and
finally lamellar sheets to the stacked lamella system in the spherical pore surface. All
these variations in the lamella morphology are results of the variation in the thickness of
the lamellar sheets with respect to the size of the pore surface in the spherical pore
geometry. The evaluation of the lamellar morphology has been investigated with respect
to the size effect in the spherical surface [8], which shows that with increasing shell
thickness, the structure gradually changes from a single lamellar sheet to two lamellae
sheets and when the film thickness is small, the perforated morphology forms in the
lamellar sheet. However, results of this study show that after the lamellar sheet and the
perforated morphology, the next destination is the stacked lamellar morphology in the
spherical pore surface.
6.2.2. Cylindrical forming system confined in spherical pores
In this section, results were obtained and discussed for cylindrical forming system
confined in spherical pore geometry with various sizes of the pore surfaces. The
cylindrical system of diblock copolymer systems were investigated in spherical pores
with neutral surfaces as well as with interacting parallel spherical interfaces, having
preferential affinity to the majority segment of the polymer system. The CDS
parameters for the cylindrical forming system were used in this study are shown in table
10.
Table 10: CDS parameter system for cylindrical forming system
f u v B D A
0.30 0.40 0.50 1.50 0.02 0.50 1.50
117
I. Cylindrical forming system in neutral spherical pores
In this section we consider cylindrical forming system confined by neutral spherical
surfaces with various pore surface sizes. Results were obtained in a pore surface with
the interior radius fixed at 3ar not shown here but in this case we get very defective
nanostructures due to the strong curvature effect in the pore surface.
In the first case, we consider the pore surface with the interior radius fixed at 5ar and
the exterior radius br which were increased by the five in each system to expand the pore
surface size.
Figure 6.7: Cylindrical forming system confined in neutral spherical pores with the interior radius of the
pore surface fixed at 5ar and the pore sizes are (a) d = 0.5 (b) d = 1 (c) d = 2.5 (d) d = 3.0.
The cylindrical forming system in the neutral spherical surfaces shows a few parallel to
the pore surface cylinders and perpendicular to the pore surface cylinders with majority.
In the pore surface with pore size d = 0.5, it shows a mixed structure of cylinders
perpendicular to the pore cylinders and cylinders parallel to the pore curved cylinders in
the pore surface as shown in Figure 6.7(a). The perpendicular to the pore surface
cylinders are in hexagonal and rhombus packing arrangements in the pore geometry. In
(a) (b)
(c) (d)
118
this pore surface a few defective spheres in the pore surface can be observed. The
defective spheres were formed due to the size effect in the pore geometry. Increasing
the pore size of the surface d = 1, defective spheres interconnect with neighbouring
cylinders to form long curved cylinders oriented parallel to the pore surface in the pore
geometry as shown in Figure 6.7(b). This pore-sized system also shows short straight
cylinders perpendicular to the pore surface. Cylinders in the spherical pore geometry
turn to be perpendicular to the pore surface with respect to the increase in the size of the
pore surface. For the pore surface having pore size d = 2.5, short straight cylinders were
induced, oriented perpendicular to the pore surface as shown in Figure 6.7(c). Notice
that in this pore size system, there are short straight cylinders perpendicular to the pore
surface on the exterior surface and long curved cylinders parallel to the pore surface on
the interior pore surface of the pore geometry. This clearly shows that long curved
cylinder parallel to the pore conforms due the curvature effect in the pore geometry. The
pore surface with pore size d = 3.0, also shows a similar story that there are short
straight cylinders perpendicular to the pore surface, while we can observe long curved
cylinders on the interior surface of the pore geometry as shown in Figure 6.7(d).
Perpendicular to the pore cylinders are in the hexagonal, square and rhombus packing
arrangements in the pore geometry.
In the second case, results were obtained and discussed with interior radius of the pore
shell were fixed at 7ar , while the exterior radius br of the pore surface were increased
to expand the pore surface.
119
Figure 6.8: Cylindrical forming system confined in neutral spherical pores with the interior radius of the
pore surface fixed at 7ar and the pore sizes are (a) d = 0.5 (b) d = 2.0 (c) d = 2.5 (d) d = 3.0.
The nanostructures were obtained by confined neutral spherical pore surfaces, with the
interior radius were fixed at 7ar , showing short straight cylinders oriented
perpendicular to the pore surface in the pore geometry. In the spherical pore system
with the pore size d = 0.5, the pore shell induces a short straight cylinder packed in
rhombus and triangular arrangements, while there are very few defective spheres in the
pore geometry as shown in Figure 6.8(a). The pore surface d = 2.0 also induces short
straight cylinders oriented perpendicular to the pore surface as shown in Figure 6.8(b).
In this size of pore surface, there are a few tilted short cylinders and curved cylinders
parallel to the pore surface. The pore systems d = 2.5 and d = 3.0 also display short
cylinders with a few curved cylinders parallel to the pore surface in the pore geometry
as shown in Figure 6.8(c,d). Results in the spherical pore geometry show that under the
influence of the curvature of the pore geometry cylinders tend to form curved cylinders,
while, when increasing the size of the pore surface cylinders reorient and become
(a) (b)
(c) (d)
120
straight short cylinders perpendicular to the pore surface in the pore geometry. Straight
short cylinders are hexagonally packed in the spherical pore geometry. Parallel to the
pore cylinders have defects in order to accommodate themselves among the
perpendicular cylinders and curvature influence. The results of this study are consistent
with the experimental results [68] which were obtained in the spherical pore surfaces,
where perpendicular, hexagonally packed and parallel to the pore surface cylinders are
observed. Hexagonally packed perpendicular cylinders were also obtained in the
experimental work [65]. Parallel and perpendicular cylinders confined in spherical pores
were also obtained by the CDS method in Cartesian coordinate system using symmetric
boundary conditions on the parallel interfaces of the spherical pore [8]. In this work
Pinna et al also performed scanning force microscopy experiments on thin films floated
on the half-spherical shell on a silicon wafer and obtained the perpendicular and parallel
cylinders which are consistent with the results of this study see Figure 9(f) there in.
II. Cylinder forming system with interfacial surfaces
In this section, we consider the cylindrical forming system confined by parallel
spherical interfaces having preferential affinity with the majority segment of the
polymer system in the spherical pores. Results were obtained by the applying
preferential interacting interfaces having a strength of interaction 2.0 to the majority
block of the polymer system in the pore surface.
In the first case, results were obtained and discussed for the pore surface with the
interior radius fixed at 5ar and the pore thickness is increased by the exterior radius of
the spherical surface.
121
Figure 6.9: Cylinder system confined by parallel spherical interfaces in a spherical pore with the interior
radius fixed at 5ar and the pore surface sizes are (a) d = 1 (b) d = 1.5 (c) d = 2.5.
The cylindrical forming system confined between the parallel spherical surface walls in
a pore system with the interior radius fixed at 5ar , shows mixed morphologies of
perforated holes and long curved cylinder in the spherical pore surface. The spherical
pore surface with pore size d = 1, shows mixed patterns of perforated holes and parallel
to the pore cylinder in the single-layered spherical pore surface as shown in Figure
6.9(a). The perforated holes induced in the pore surface have rhombus, curved lines,
straight lines and L-shaped packing arrangements on the pore surface. Perforated holes
morphology was also obtained by the experimental work [31] shown in Figure 6(a)
there in, which is consistent with the results shown in Figure 6.9(a). The coexistence of
the cylinders parallel to the pore and perforated holes were also reported in the planar
geometry [93]. The coexistence of both morphologies were also verified by Pinna et al
on the surface of the sphere both computationally and experimentally [8] while their
experimental image (shown in Figure 10(d) there in) shows consistency with this study.
However, the pore surface with pore size d = 1.5 shows bi-layered structure and induces
(a) (b)
(c)
122
long curved cylinders, short straight cylinders, a few perforated holes and defective
spheres in the pore surface as shown in Figure 6.9(b). Both layers are interconnected by
cross-sectional cylinders, as one can observe by exterior view of the interior layer of the
nanostructure shown on the right hand side of the image shown in Figure 6.9(b). The
pore surface with pore size d = 2.5 a tri-interconnected layered nanostructure and shows
long curved cylinders, short straight cylinders oriented perpendicular to the pore surface
through cross-sections of the pore surface, as shown in Figure 6.9(c). There are a few
defective spheres observed in the pore surface while there are very few perforated holes
in the nanostructure.
Parallel to the pore cylinders were also obtained by both experimental and
computational work [8] and their obtained results were shown by Figure 9(a) and Figure
9(c) there in, are consistent with this study. The coexistence of parallel, perpendicular
and perforated morphology of cylindrical forming system were also reported in the
computational study of the planar cylinders [92]. The cylinders parallel to the pore
surface in the spherical geometry were reported in the experimental work [31] see
Figure 6(c) there in.
In the next, we consider the pore surface with interior radius fixed at 7ar , while the
exterior radius br of the pore surface was increased to expand the pore surface and pore
surface walls have affinity with the majority block of the polymer system.
123
Figure 6.10: Cylinder system confined by parallel spherical interfaces in a spherical pore with the interior
radius fixed at 7ar and the pore surface sizes are (a) d = 1 (b) d = 1.5 (c) d = 2.0.
Results in the pore surface with the interior radius fixed at 7ar and confined by
parallel spherical surfaces having preferential affinity to the major segment of the
polymer system show long curved cylinders, short straight cylinders and defective
spheres in the spherical shells. The first system with narrow pore size d = 1 shows
defective squeezed spheres in the middle of the surface, long as well as short curved
cylinders and perforated holes on the top and bottom of the spherical pore surface as
shown in Figure 6.10(a). The next pore surface with pore size d = 1.5, displays a single
layered nanostructure as shown in Figure 6.10(b). This nanostructure induces long
curved cylinders, short cylinders parallel to the pore surface and defective spheres in the
pore surface. However, this pore-sized surface-perforated morphology did not
conformed in the spherical pore surface due to the size of the system. The spherical
shell with pore size d = 2.0 is a bi-layered nanostructure which induces the long curved
cylinders, straight cylinder parallel to the pore surface, short cylinders perpendicular to
the pore surface through cross section of the pore surface and defective spheres in the
pore surface as shown in Figure 6.10(c). Similar cylindrical coils wrapping around each
(a) (b)
(c)
124
other in the surface of sphere can be found in experimental work [85], where shown by
Figure 5 therein.
6.2.3. Sphere forming system confined in spherical pores
In this section, a spherical forming system is investigated in spherical pore shell with
various pore sizes by cutting off the spherical surface from its interior pore surface.
Results are obtained with modified CDS parameters for sphere forming system, with
high temperature in the neutral pore surfaces, while, in the results obtained with
interfacial surfaces, CDS parameters for sphere forming system with low temperature
were used in the pore geometry.
I. Sphere forming system confined in neutral spherical surfaces
In this subsection, results were obtained and discussed for spherical forming systems in
the neutral spherical pore surfaces with various pore thicknesses and pore radii of the
systems. Results were obtained by the new modified CDS parameter system with high
temperature as shown in table (11).
Table 11: Modified CDS parameter system for sphere forming system with high temperature
f u v B D A
0.25 0.40 0.50 1.5 0.02 0.50 1.50
In the first case, results are obtained and discussed for a pore surface with the interior
radius fixed at 3ar , while the exterior radius br increased to expand the pore thickness
of the system. Results were obtained in the neural pore surfaces with the modified CDS
parameter system with high temperature as shown in table (11).
125
Figure 6.11: Modified sphere system in the neutral pore surfaces with the interior radius of the pore
surface fixed at 3ar and the pore sizes are (a) d = 0.5 (b) d = 1.0.
The results show that spheres are packed in the pore surface with thick defective
cylinders in the neutral pore surfaces with the interior radius of the pore surfaces was
fixed at 3ar . These defective cylinders were formed in the pore geometry by the
merger of two or more spheres due to the influence of curvature. The pore surface with
the pore size d = 0.5 shows a single-layered nanostructure with packings of the spheres
including defective thick cylinders as shown in Figure 6.11(a). However, the pore
surface having pore size d = 1 show defective thick cylinders on its interior surface
only, while the exterior surface shows sphere packing in the pore surface as shown in
Figure 6.11(b). Results show that defective cylinders form because of the strong
curvature influence in the pore geometry.
In the second case, we consider the pore surface with the interior radius of the pore
surface fixed at 5ar and the exterior radius br were varied to obtained different pore
thicknesses of the spherical pore shells. Results were obtained in the neutral spherical
pore surfaces with modified CDS parameter system with high temperature.
(a) (b)
126
Figure 6.12: Modified sphere system in the neutral pore surfaces with the interior radius of the pore
surface fixed at 5ar and the pore sizes are (a) d = 0.5 (b) d = 1.0 (c) d = 1.5.
Results in the neutral spherical pores with the interior radius fixed at 5ar were
obtained with modified CDS parameter system for sphere forming system show packing
arrangements of sphere system in the pore geometry, such as packing along the vertices
of the concentric squires, packing along the quadrilateral vertices, square packing,
pentagon packing, hexagonal packing and rhombus packing inside the pore surface. The
narrow spherical pore system d = 0.5, shows very interesting packings of the sphere
system in the spherical pore geometry as shown in Figure 6.12(a). This pore surface
induces spheres in the packing arrangement along the vertices of the concentric squares
in the pore geometry. The spheres in the outer layer (cut off in half) are shifted with
respect to the spheres in the interior layer so that the sphere from the interior layer
emanates in the middle of the squared-packed spheres of the outer layer. It is interesting
that the sphere system on the curved surface is still able to persist in the square packing
arrangement. Increasing the pore thickness of the pore surface d = 1, we can observe
square packing, rhombus packing and pentagon packing of the sphere system on the
(a)
(b) (c)
127
pore surface as shown in Figure 6.12(b). However, in this pore surface packing along
the vertices of the concentric squares and quadrilaterals did not conform to the pore
geometry, due to the size effect. In the pore surface d = 1.5, spheres are induced square,
rhombus, pentagon, and hexagonal arrangements in the pore geometry as shown in
Figure 6.12(c).
In the second case we consider the pore surface with interior radius of the pore system
fixed at 7ar and the pore surface size was extended by the exterior radius br of the
spherical shells.
Figure 6.13: Sphere system obtained by modified CDS parameter system in the neutral pore surfaces with
the interior radius of the pore surface fixed at 7ar and the pore sizes are (a) d = 0.5 (b) d = 1.0 (c) d =
1.5.
The modified sphere system obtained in a pore surface with interior radius fixed at
7ar show square packing and rhombus packing arrangements in the spherical pore
geometry. The pore surface with pore size d = 0.5, shows square and rhombus packing
arrangements of the spheres in the spherical shell as shown in Figure 6.13(a). Body-
centred-cubic nanodomains were packed in the concentric layers of nano structure in
(a) (b)
(c)
128
their intrinsic packing arrangements in both layers of the nano structure. The interior
layer of the spherical pore shell was shifted so that the nanodomains in the interior layer
emanates from the centre of four nano domains of the outer layer in the pore structure.
The pore surface with pore size d =1 also induces sphere system in square and rhombus
arrangements in the pore surface as shown in Figure 6.13(b). In this pore surface sphere
were conformed in distorted concentric square arrangements which can be observed in
Figure 6.13(b) shown by strips on the image. Results show more shifting of the interior
layer with respect to the exterior layer of the microstructure due to the size effect. The
pore surface having pore size of the system d =1.5, shows more distorted square and
rhombus packing arrangements of the nanodomains on account of the internal energy of
the polymers system as shown in Figure 6.13(c).
II. Sphere system with interfacial surfaces
In this subsection, the sphere system is investigated by confining interacting parallel
spherical surfaces having preferential affinity with the majority block of the polymer
system in the spherical shell surfaces with various thicknesses of the pore systems.
Interaction strength between the surface walls and the majority segment of the polymer
system applied is 2.0 . CDS parameters for sphere forming system with low
temperature were used in the simulation results, which are shown in table (12).
Table 12: CDS parameters for sphere forming system with low temperature
f u v B D A
0.20 0.40 0.50 2.30 0.01 0.50 1.50
In the first case, we consider the spherical pore shell with the interior radius fixed at
5ar and the exterior radius br varied to expand the thicknesses of the pore shells.
129
Figure 6.14: Sphere system confined by parallel to the pore walls, with the interior radius of the pore
surface fixed at 5ar and the pore surface sizes are (a) d = 1.5 (b) d = 2.0 (c) d = 2.5.
Results obtained in the spherical pore shell under confinement of the parallel spherical
interfaces show square and rhombus packing arrangements along with the defective
short cylinders in the pore geometry. The pore shell with pore size 1.5, induces nano
domains in a single-layered nano structure with square and rhombus packing
arrangements in the spherical pore shell as shown in Figure 6.14(a). Some of the nano
domains preserved their classical pentagonal and hexagonal conformation in the pore
shell under the influence of the confinement and curvature. However, some of the
nanodomains are packed in the square and rhombus arrangements in the spherical pore
surface. Rhombus packing arrangement of the sphere system forms which is due to the
curved surface effect in the pore geometry and originally this packing arrangement is
the square packing arrangement of the sphere system. There are short curved cylinders
and straight short cylinders parallel to the pore surface in the pore shell. The defective
cylinders form due to the merger of two or more spheres. In the pore shell with size 2.0
spheres are induced in a bi-layered nanostructure with the interior layer shifted with
respect to the outer layer of the nanostructure as shown in Figure 6.14(b). The spheres
(a) (b)
(c)
130
in the pore surface show skewed hexagonal, pentagonal and rhombus formations with
few defective short cylinders in the pore surface. Short defective cylinders are parallel
to the pore surface and perpendicular to the pore surface (see in the interior layer on the
right hand side of the system) while there are more defective cylinders in the interior
layer as compared to the outer layer of the nanostructure which the role of the curvature
in the formation of the defective short cylinder in the pore geometry. The pore shell
having pore size d =2.5 also induces sphere system in a single layered nanostructure
with less defective cylinders as compared to previously-sized systems as shown in
Figure 6.14(c). This system also shows twisted hexagonal, pentagonal and squared
arrangements of the nanodomains in the pore shell. The less defective spheres in the
pore shell show that the size effect is also one of the key reasons behind the formation
of the defective cylinders in the pore geometry.
In the second case, we consider a pore shell with the interior radius fixed at 7ar and
the pore shell size was increased by the exterior radius br of the pore shell. The results
were obtained with the surface preference to the majority segment of the polymer
system in the pore geometry.
131
Figure 6.15: Sphere system confined by parallel to the pore walls, with the interior radius of the pore
surface fixed at 7ar and the pore surface sizes are (a) d = 1.5 (b) d = 2.0 (c) d = 2.5.
Nanodomains in the spherical pore shell with the interior radius fixed at 7ar confined
by parallel spherical interfaces show more skewed hexagonal, pentagonal and square
packing arrangements in the pore geometry as compared to previous pore shell systems,
while in this pore system a less defective system can be observed, nanodomains are
small in size. The pore shell with pore size d = 1.5 shows a single-layered nanostructure
with skewed pentagonal, square and rhombus packing arrangement of the micro
domains as shown in Figure 6.15(a). The pore shell also induces the short curved and
straight cylinder oriented parallel to the pore shell. The pore shell having pore size d
=2.0 also show a single-layered nanostructure with square and rhombus packing
arrangements of the nanodomains in the pore geometry as shown in Figure 6.15(b).
There are skewed nanodomains in the respect of the size and shape as well in the pore
surface. Defective cylinders in the pore shell are short straight cylinders parallel to the
pore surface and one arc-shaped cylinder evident on the pore surface. In the pore
surface with size of the pore d = 2.5 displays spheres in the bi-layered nanostructure as
shown in Figure 6.15(c.) This pore shell also shows the skewed pentagonal, square, and
(a) (b)
(c)
132
rhombus packing arrangements in the both layers of the nanostructure in the pore
geometry. Defective cylinders can be seen on the surface of the outer layer and the
interior layer of nanostructure. There are a few straight cylinders perpendicular to the
pore shell which can be seen clearly in the interior layer of the pore shell surface.
However, in this pore system spheres are not skewed in respect of their size and shape
in the nanostructure.
6.3. Summary
Lamella systems in neutral spherical pore shells show, concentric lamella (onion-like
structure), standing lamella, stacked lamella and perforated lamella morphologies in the
pore geometry. In the absence of the interfacial surfaces, results show that the stacked
lamella morphology conformed in the pore geometry with respect to the pore size. In
the presence of the interfacial surfaces, The obtained results predict the coexistence of
lamella, cylinder and sphere morphologies in the pore geometry. Interfacial spherical
surfaces also show the conformation of the stacked lamellar system with respect to the
size of the pore surface. The confinement-induced lamella evolves in patterns such as
concentric lamella, perforated lamella, mixed concentric lamella and stacked lamella. In
the case of confined cylinders with neutral surfaces, cylinders parallel to the pore and
perpendicular to the pore cylinders observed in the pore surfaces. The confinement
induces cylinders perpendicular to the pore surface with respect to the pore size. In the
presence of the interfacial pore walls, perforated holes and parallel to the pore cylinders
observed in the pore geometry. The conformations of the short cylinders perpendicular
to the pore surface were observed with respect to the pore size of the surface. Long
curved cylinders, short cylinders and perforated holes were observed in the
nanostructure in the concentric layers. Pore geometry induces the spheres in square,
rhombus, pentagon, hexagon packing arrangements in the absence of the interfacial pore
133
walls. However, these packing arrangements of the sphere forming system were
distorted with respect to the size of the system. In the presence of the interfacial
surfaces, spheres packed in the concentric layers were observed in the pore geometry.
The square and rhombus packing arrangements in the presence of interacting surfaces
was observed in confined nanostructures.
134
CHAPTER SEVEN
7. Conclusions and Future work
7.1. Block copolymers confined in the circular annular pores
Using the CDS method employed in the polar coordinate system we obtained novel
nanostructures in the circular pore geometry for lamella, cylinder and sphere forming
systems. The asymmetric lamella forming system in the neutral film interfaces shows
perpendicular to the exterior circumference and parallel to the interior circumference
tendency in the circular pore system. Grain boundaries confined in the circular pore
geometry show various formations of lamella including Y-shape, U-shape, W-shape, V-
shape and T-junctions. Fingerprint morphology confined in the neutral circular pore
with various pore thicknesses induced clusters of perforated holes, isolated perforated
holes and star lamella with perforated hole at the centre of star. Close to the exterior
pore boundary, lamellae conforms concentric parabolic patterns having openings in the
direction of outer boundary, however for larger system sizes these parabolic patterns
evolve into the parallel lamella strips normal to the outer circular boundary. The pore
system with large interior radii shows parallel strips lamellae normal to the circular pore
boundaries. The interplay of both interfacial circular walls and curvature influence
induces lamella into the concentric circular rings in the circular pore geometry. In the
presence of interfacial circular walls, concentric alternating lamella (onion-like)
nanostructure were obtained in the circular pores which are strongly consistent with
experimental results. Results show that under the influence of curvature and interfacial
circular walls, the microdomains tend to form concentric lamella in the pore geometry.
Symmetric lamella system induces defective nanostructures in the circular pore
geometry.
135
The cylindrical forming system in the neutral circular annular pore system shows, the
novel packing arrangements along the spiral lines. However, in the larger pore systems
spiral packing arrangements alters and system regains in the classical hexagonal
packing arrangements in the circular pore system. While, in the presence of interacting
circular walls microdomains induces packing arrangements in the concentric circular
rings in the pore geometry.
Similarly, the sphere forming system confined in circular annular pores shows, packing
arrangements along the spiral lines, parabolic lines with opening along the outer
boundary and hexagonal packing arrangements in the pore system. A new set of CDS
parameters for sphere forming system were also introduced in the study. Results
obtained with modified CDS parameters show that small system size induces sphere
along the spiral curve, while large circular pore system induces sphere in the classical
hexagonal packing arrangements. In the presence of interfacial circular walls, sphere
system with modified CDS parameters squeezed down in size and shape. Sphere system
under geometric confinement of annular circular pores with interfacial circular walls
show, packing arrangements in the concentric circular rings. Results shows that under
the influence of the curvature nanodomains show spiral packing arrangements while due
to the interplay of both curvature and interfacial circular walls microdomains forms
packing arrangements in the concentric circular rings in the pore geometry
7.2. Block copolymers confined in cylindrical pores
The lamella forming system confined in the neutral cylindrical pores show mixed
orientations like helical strip lamella, various formations of strip lamella and lamellar
sheets parallel to the pore axis. In the small pore radius, helical strip lamella conforms
in the neutral cylindrical pores, while in the large pore radius lamellar sheets conforms
in the cylindrical pore geometry. Results show that curvature influences the lamella
136
system to form helical strips around the pore axis while, in the minimal curvature effect
lamella tend to form lamellar sheets parallel to the pore axis. In the presence of the
parallel to the pore interfacial surfaces, the pore system induces smooth concentric
cylindrical layers and the numbers of concentric layers depend on the size of the system.
However, in the presence of the perpendicular to the pore interfacial surfaces,
cylindrical geometry induces lamella into the smooth cylindrical lids perpendicular to
the pore surface. In this case, the system shows strong segregation limit in the narrow
sized surfaces under perpendicular one dimensional confinement. In the presence of
both parallel and perpendicular interfacial surfaces, cylindrical confinement induces
lamella into a single circular ring cylinder.
The cylinder forming system under geometric confinements of the neutral cylindrical
pore show mixed orientations of microdomains, perpendicular to pore surface and
parallel to the pore surface. Cylinders perpendicular to the pore were packed in square,
rhombus and hexagonal arrangements in the neutral pore geometry. In the neutral pore
surfaces, system induces short straight cylinders perpendicular to the pore surface and
curved cylinders parallel to the pore cylinders in the pore geometry. However, in the
presence of parallel interfacial surfaces with small pore radius, microdomains show
parallel to the pore surface orientation and wrapped around the pore system. In the
presence of the parallel interfacial surfaces cylinders were packed in the concentric
cylindrical layers.
In the narrow pore size system, coexistence of perforated morphology and parallel to the
pore cylinders were observed in the pore geometry. Results show that the interplay of
both curvature and interfacial parallel cylindrical surfaces alter the orientation of the
perpendicular cylinders to the parallel coil cylinders wrapped around the cylindrical
axis. In the large pore radius with parallel interfacial surfaces, cylinders show mixed
orientation in the pore geometry. In the short length pore surface, we obtained standing
137
cylinders which were packed in the concentric circles in the pore system. Cylindrical
forming systems were also investigated with various lengths of the narrow sized
cylindrical pores, with one dimensional confinement (parallel interfacial surfaces) and
two dimensional confinements (parallel and perpendicular interfaces). In this case
coexistence of perforated holes with parallel to the pore cylinders were observed,
however, under two dimensional confinements in the small pore length, cylindrical
rings arranged perpendicularly one on other were observed.
The sphere forming system obtained in neutral cylindrical pores with CDS parameter
system with low temperature show, sphere packing along with defective short and long
curved cylinders perpendicular to the pore surface. The sphere forming systems were
obtained with a new set of CDS parameter system with high temperature. In this case
we obtained perfect sphere systems in the cylindrical pores without any defective
cylinders. The sphere system obtained with modified CDS parameters and with high
temperature show, packing arrangements along the perimeters of the concentric
quadrilateral in the narrow sized pore surface. This packing arrangements look like
diagonal lines on the cylindrical pore surface. The modified sphere system under
preferential surfaces squeezed down in size and shape in the pore surface. Sphere
forming system confined in the cylindrical pore with interfacial surfaces show, packing
arrangements into concentric circular layers. In the presence of the perpendicular and
parallel interfacial surfaces, pore geometry induces spheres into the circular rings
arranged in parallel and perpendicular layers. In this case we obtained in small sized and
short length cylindrical pore, the coexistence of spheres and cylinder segments induced
into a circular ring. Results show that cylindrical pore geometry induces spheres into the
layers of circular rings.
138
7.3. Block copolymers confined in spherical pores
Lamella systems in neutral spherical pore shells show, concentric lamella (onion like
structure), standing lamella, stacked lamella and perforated lamella morphologies in the
pore geometry. In the absence of the interfacial surfaces, results show that the stacked
lamella morphology conformed in the pore geometry with respect to the pore size. In
the presence of the interfacial surfaces, we obtained the nanostructure which predicted
the coexistence of lamella, cylinder and sphere morphologies in the pore geometry.
Interfacial spherical surfaces also show the conformation of the stacked lamellar system
with respect to the size of the pore surface. The confinement induced lamella evolves in
the patterns like concentric lamella, perforated lamella, mixed concentric lamella and
stacked lamella. In the case of confined cylinders with neutral surfaces, parallel to the
pore cylinders and perpendicular to the pore cylinders observed in the pore surfaces.
The confinement induces cylinders perpendicular to the pore surface with respect to the
pore size. In the presence of the interfacial pore walls, perforated holes and parallel to
the pore cylinders observed in the pore geometry. The conformations of the short
cylinders perpendicular to the pore surface were observed with respect to the pore size
of the surface. The long curved cylinders, short cylinders and perforated holes were
observed in the nano structure in the concentric layers. Pore geometry induces the
spheres in square, rhombus, pentagon, hexagon packing arrangements in the absence of
the interfacial pore walls. However, these packing arrangements of the sphere forming
system distorted with respect to the size of the system. In the presence of the interfacial
surfaces, spheres packed in the concentric layers were observed in the pore geometry.
The square and rhombus packing arrangements in the presence of interacting surfaces
was observed in confined nanostructures.
139
7.4. Future work
For future work following projects can be carried out.
I. To study Block copolymers confinement’s in circular pores, cylindrical pores
and spherical pores by applying electric field and shear flow using the CDS
method employed in curvilinear coordinate systems.
II. To study and investigate block copolymer confined in circular pores, cylindrical
pores and spherical pores by using asymmetric boundary conditions on the pore
interfaces in the circular pores, cylindrical pores and spherical pores.
III. To study block copolymer under various different confinements, in addition to
the circular pore, cylindrical pore and spherical pore geometries, study will also
be carried out under geometric confinements of ellipsoid and hyperboloid.
IV. The study and investigate block copolymers using two order parameters for
diblock copolymers/homopolymer mixture confined in the circular pores,
cylindrical pores and spherical pores using physically motivated discretization.
140
References
[1] M. Folkes, Processing, Structure and Properties of Block Copolymers. Springer
Science & Business Media, 2012.
[2] P. Wu, G. Ren, C. Li, R. Mezzenga and S. A. Jenekhe, "Crystalline diblock
conjugated copolymers: Synthesis, self-assembly, and microphase separation of poly (3-
butylthiophene)-b-poly (3-octylthiophene)," Macromolecules, vol. 42, pp. 2317-2320,
2009.
[3] S. An-Chang and L. Baohui, "self-assembly of diblock copolymers under
confinement," RSC, vol. 9, pp. 1398-1413, 2013.
[4] H. Xiang, K. Shin, T. Kim, S. I. Moon, T. J. McCarthy and T. P. Russell, "From
cylinders to helices upon confinement," Macromolecules, vol. 38, pp. 1055-1056, 2005.
[5] K. Shin, S. Obukhov, J. Chen, J. Huh, Y. Hwang, S. Mok, P. Dobriyal, P.
Thiyagarajan and T. P. Russell, "Enhanced mobility of confined polymers," Nat Mater,
vol. 6, pp. 961-965, print, 2007.
[6] Q. Wang, "Symmetric diblock copolymers in nanopores: Monte Carlo simulations
and strong-streching theory," J. Chem. Phys., vol. 126, pp. 024903-1-024903-11, 2007.
[7] P. Chen, X. He and H. Liang, "Effect of surface field on the morphology of a
symmetric diblock copolymer under cylindrical confinement," J. Chem. Phys., vol. 124,
pp. 104906-1-104906-6, 2006.
[8] M. Pinna, S. Hiltl, X. Guo, A. Boker and A. V. Zvelindovsky, "Block Copolymer
Nanocontainers ," ACSNANO, vol. 4, pp. 2845-2855, 2010.
141
[9] J. Feng, H. Liu and Y. Hu, "Mesophase seperation of diblock copolymer confined in
a cylindrical tube studeid by dissipative particle dynamics," Macromol, Theory Simul,
vol. 15, pp. 674-685, 2006.
[10] P. Chen and H. Liang, "Origin of microstructures from confined asymmetric
diblock copolymers," Macromolecules, vol. 40, pp. 7329-7335, 2007.
[11] M. Yiyong and E. Adi, "Self-assembly of diblock copolymers," Chem Soc Rev, vol.
41, pp. 5969-5985, 2012.
[12] M. Pinna and A. V. Zvelindovsky, "Large scale simulation of block copolymers
with cell dynamics," Eur. Phys. J. B, vol. 85, pp. 210, JUN, 2012.
[13] J. Feng and E. Ruckenstein, "Morphologies of AB diblock copolymer melt
confined in nanocylindrical tubes," Macromolecules, vol. 39, pp. 4899-4906, 2006.
[14] b. Yu, P. Sun, T. Chen, Q. Jin, D. Ding, B. Li and A. Shi, "Confinement-induced
novel morphologies of block copolomers," PRL, vol. 96, pp. 138306-1-138306-4, 2006.
[15] K. Shin, H. Xiang, S. I. Moon, T. Kim, T. J. McCarthy and T. P. Russell. Curving
and frustrating flatland. Science 306(5693), pp. 76-76. 2004. . DOI:
10.1126/science.1100090.
[16] Y. Wu, G. Cheng, K. Katsov, s. w. Sides, J. Wang, J. Tang, G. H. Fredrickson, M.
Moskovits and G. D. Stucky, "Composite mesostructures by nano-confinement," Nature
Publishing Group, vol. 3, pp. 816-822, 2004.
[17] G. J. A. Sevink, A. V. Zvelindovsky and Fraaije, J. G. E. M, "Morphology of
symetric block copolymer in a cylindrical pore," Journal of Chemical Physics, vol. 115,
pp. 8226-8230, 2001.
142
[18] C. Lam, "Applied Numerical Methods for partial differential equations," Prentice
Hall, 1994.
[19] R. Yang, B. Li and A. Shi, "Phase behaviour of binary blends of diblock
copolymer/homopolymer confined in spherical nanopores," Langmuir, vol. 28, pp.
1569-1578, 2012.
[20] M. Pinna, X. Guo and A. V. Zvelindovsky, "Block copolymer nanoshells,"
Polymer, vol. 49, pp. 2797-2800, 2008.
[21] B. Yu, P. Sun, T. Chen, Q. Jin, D. Ding, B. Li and A. Shi, "Self-assembly of
diblock copolymers confined in cylindrical nanopores," J. Chem. Phys., vol. 127, pp.
114906-1-114906-15, 2007.
[22] T. Chantawansri L., A. W. Boss, A. Hexemer, H. D. Ceniceros, c. J. Garcia-
cervera, E. J. Kramer and G. H. Fredrickson, "Self-Consistent field theory simulations
of block copolymer assembly on a sphere," Physical Review E, vol. 75, pp. 031802-1-
031802-17, 2007.
[23] X. He, H. Liang, M. Song and C. Pan, "Possiblity of design of Nanodevices by
confined macromolecular self-assembly," Macromol, Theory Simul, vol. 11, pp. 379-
382, 2002.
[24] W. Li and R. A. Wickham, "Self-Assembled Morphologies of a diblock copolymer
melt confined in a cylindrical nanopore," Macromolecules, vol. 39, pp. 8492-8498,
2006.
[25] M. W. Matsen and F. S. Bates, "Unifying Weak- and Strong-Segregation Block
Copolymer Theories," Macromolecules, vol. 29, pp. 1091-1098, 01/01, 1996.
143
[26] F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and
experiment," Annu. Rev. Phys. Chem., vol. 41, pp. 525-557, 1990.
[27] N. Hadjichristidis, A. Hirao, Y. Tezuka and F. Du Prez, Complex Macromolecular
Architectures: Synthesis, Characterization, and Self-Assembly. John Wiley & Sons,
2011.
[28] M. Pinna, Mesoscale Modelling of Block Copolymer Systems. Germany: VDM
Verlag, 2010.
[29] C. M. M. a. M. E. Cates, "Harmonic Corrections near the Ordering Transition,"
EPL (Europhysics Letters), vol. 13, pp. 267, 1990.
[30] S. R. Ren and I. W. Hamley, "Cell Dynamics Simulations of Microphase
seperation in block copolymers," Macromolecules, vol. 34, pp. 116-126, 2001.
[31] D. A. Rider, J. I. L. Chen, C. Eloi, A. C. Arsenault, T. P. Russell, Ozin G. A and I.
Manners, "Controlling the Morphologies of Organometallic Block Copolymers in the 3-
Dimensional Spatial Confinement of Colloidal and Inverse Colloidal Crystals,"
Macromolecules, vol. 41, pp. 2250-2259, 2008.
[32] P. Dobriyal, H. Xiang, M. Kazuyuki, J. Chen, H. Jinnai and T. P. Russell,
"Cylindrically Confined Diblock Copolymers," Macromolecules, vol. 42, pp. 9082-
9088, 11/24, 2009.
[33] A. K. Khandpur, S. Foerster, F. S. Bates, I. W. Hamley, A. J. Ryan, W. Bras, K.
Almdal and K. Mortensen, "Polyisoprene-polystyrene diblock copolymer phase diagram
near the order-disorder transition," Macromolecules, vol. 28, pp. 8796-8806, 1995.
144
[34] G. H. Fredrickson, "Surface ordering phenomena in block copolymer melts,"
Macromolecules, vol. 20, pp. 2535-2542, 10/01, 1987.
[35] R. Choksi and X. Ren, "On the Derivation of a Density Functional Theory for
Microphase Separation of Diblock Copolymers," Journal of Statistical Physics, vol.
113, pp. 151-176, 2003.
[36] C. M. M. a. M. E. Cates, "Harmonic Corrections near the Ordering Transition,"
EPL (Europhysics Letters), vol. 13, pp. 267, 1990.
[37] I. W. Hamley, Introduction to Soft Matter. England: John Wiley & Sons, Ltd,
2007.
[38] S. Kim, W. Li, G. H. Fredrickson, C. J. Hawker and E. J. Kramer, "Order–disorder
transition in thin films of horizontally-oriented cylinder-forming block copolymers:
thermal fluctuations vs. preferential wetting," Soft Matter, 2016.
[39] F. H. Schacher, P. A. Rupar and I. Manners, "Functional block copolymers:
nanostructured materials with emerging applications," Angewandte Chemie
International Edition, vol. 51, pp. 7898-7921, 2012.
[40] W. Li, R. A. Wickham and R. A. Garbary, "Phase diagram for a diblock copolymer
melt under cylindrical confinement," Macromolecules, vol. 39, pp. 806-811, 2006.
[41] Y. Mai and A. Eisenberg, "Self-assembly of block copolymers," Chem. Soc. Rev.,
vol. 41, pp. 5969-5985, 2012.
[42] V. Bart, U. K. Jaeup, L. C. Tanya, H. F. Glenn and W. M. Mark, "Self-Consistent
field theory for diblock copolymers grafted to a sphere," Soft Matter, vol. 7, pp. 5049-
5452, 2011.
145
[43] B. Yu, B. Li, Q. Jin, D. Ding and A. Shi, "Confined Self-Assembly of Cylinder-
forming diblock copolymer: effects of confing geometries," softMatter, RSC Publishing,
vol. 7, pp. 10227-10240, 2011.
[44] J. G. Son, J. Chang, K. K. Berggren and C. A. Ross, "Assembly of sub-10-nm
block copolymer patterns with mixed morphology and period using electron irradiation
and solvent annealing," Nano Letters, vol. 11, pp. 5079-5084, 2011.
[45] I. Hamley, "Nanostructure fabrication using block copolymers," Nanotechnology,
vol. 14, pp. R39, 2003.
[46] M. Ma, E. L. Thomas, G. C. Rutledge, B. Yu, B. Li, Q. Jin, D. Ding and A. Shi,
"Gyroid-forming diblock copolymers confined in cylindrical geometry: A case of
extreme makeover for domain morphology," Macromolecules, vol. 43, pp. 3061-3071,
2010.
[47] V. A. J.M.G. Cowie, Polymers: Chemistry and Physics of Modern Materials. CRC
Press, 2008.
[48] D. E. Angelescu, J. H. Waller, R. A. Register and P. M. Chaikin, "Shear‐Induced
Alignment in Thin Films of Spherical Nanodomains," Adv Mater, vol. 17, pp. 1878-
1881, 2005.
[49] M. Matsen, "Electric field alignment in thin films of cylinder-forming diblock
copolymer," Macromolecules, vol. 39, pp. 5512-5520, 2006.
[50] S. Pujari, R. A. Register and P. M. Chaikin, "Shear induced alignment of standing
lamellar block copolymer thin films," in 2011 AIChE Annual Meeting, 11AIChE, 2011, .
146
[51] J. N. L. Albert and T. H. Epps III, "Self-assembly of block copolymer thin films,"
Materials Today, vol. 13, pp. 24-33, 6, 2010.
[52] X. Wang, S. Li, P. Chen, L. Zhang and H. Liang, "Microstructures of lamella-
forming diblock copolymer melts under nanorod-array confinements," Polymer, vol. 50,
pp. 4964-4972, 2009.
[53] A. Daud Júnior, F. M. Morais, S. Martins, D. Coimbra and W. A. Morgado,
"Microphase separation of diblock copolymer with moving walls," Brazilian Journal of
Physics, vol. 34, pp. 405-407, 2004.
[54] V. Weith, A. Krekhov and W. Zimmermann, "Stability and orientation of lamellae
in diblock copolymer films," J. Chem. Phys., vol. 139, pp. 054908, 2013.
[55] I. W. Hamley, "Cell dynamics simulations of block copolymers," Macromolecular
Theory and Simulations, vol. 9, pp. 363-380, 2000.
[56] S. Pujari, M. A. Keaton, P. M. Chaikin and R. A. Register, "Alignment of
perpendicular lamellae in block copolymer thin films by shearing," Soft Matter, vol. 08,
pp. 5358-5363, 2012.
[57] N. E. Voicu, S. Ludwigs and U. Steiner, "Alignment of Lamellar Block
Copolymers via Electrohydrodynamic‐Driven Micropatterning," Adv Mater, vol. 20, pp.
3022-3027, 2008.
[58] G. J. A. Sevink and A. V. Zvelindovsky, "Block copolymers confined in a
nanopores: Pathfinding in a curveing and frustrating flatland," Journal of Chemical
Physics, vol. 128, pp. 084901-1-084901-16, 2008.
147
[59] M. Pinna, X. Guo and A. V. Zvelindovsky, "Diblock copolymers in a cylindrical
pore," J. Chem. Phys., vol. 131, pp. 214902-1-214902-7, 2009.
[60] M. W. Matsen and M. Schick, "Stable and Unstable Phases of a Diblock
Copolymer Melt," Physical Review Letters, vol. 72, pp. 2660-2663, 1994.
[61] P. Chen, H. Liang and A. Shi, "Microstructures of a cylinder-forming diblock
copolymer under spherical confinement," Macromolecules, vol. 41, pp. 8938-8943,
2008.
[62] P. Chi, Z. Wang, B. Li and A. Shi, "Soft confinement-induced morphologies of
diblock copolymers," Langmuir, vol. 27, pp. 11683-11689, 2011.
[63] P. Broz, S. Driamov, J. Ziegler, N. Ben-Haim, S. Marsch, W. Meier and P.
Hunziker, "Toward Intelligent Nanosize Bioreactors: A pH-Switchable, Channel-
Equipped, Functional Polymer Nanocontainer," Nano Lett., vol. 6, pp. 2349-2353,
10/01, 2006.
[64] W. Chen, H. Wei, S. Li, J. Feng, J. Nie, X. Zhang and R. Zhuo, "Fabrication of
star-shaped, thermo-sensitive poly(N-isopropylacrylamide)–cholic acid–poly(ɛ-
caprolactone) copolymers and their self-assembled micelles as drug carriers," Polymer,
vol. 49, pp. 3965-3972, 8/26, 2008.
[65] A. Knoll, A. Horvat, K. S. Lyakhova, G. Krausch, G. J. A. Sevink, R. Magerle and
R. Magerle, "Phase Behavior in Thin Films of Cylinder-Forming Block Copolymers,"
Physical Review Letters, vol. 89, pp. 035501-1, 2002.
[66] A. Knoll, R. Magerle and G. Krausch. Phase behavior in thin films of cylinder-
forming ABA block copolymers: Experiments. J. Chem. Phys. 120(2), pp. 1105-1116.
2004. . DOI: http://dx.doi.org/10.1063/1.1627324.
148
[67] A. Knoll, K. S. Lyakhova, A. Horvat, G. Krausch, G. J. A. Sevink, A. V.
Zvelindovsky and R. Magerle, "Direct imaging and mesoscale modelling of phase
transitions in a nanostructured fluid," Nat Mater, vol. 3, pp. 886-891, print, 2004.
[68] H. Takeshi, T. Atsunori, Y. Hiroshi and S. Masatsugu, "Spontaneous formation of
polymer nanoparticles with inner micro-phase separation structures," Soft Matter, vol. 4,
pp. 1302-1305, 2008.
[69] P. Tang, F. Qiu, H. Zhang and Y. Yang, "Phase seperation patterns for diblock
copolymers on spherical surfaces: A finite volume method," Physical Review E, vol. 72,
pp. 016710-1-016710-7, 2005.
[70] W. Li, R. A. Wickham and R. A. Garbary, "Phase Diagram for a Diblock
Copolymer Melt under Cylindrical Confinement," Macromolecules, vol. 39, pp. 806-
811, 01/01, 2006.
[71] G. H. Griffths, B. Vorselaars and M. W. Matsen, "Unit Cell Approximation for
diblock copolymer brushes grafted to spherical particles," Macromolecules, vol. 44, pp.
3649-3655, 2011.
[72] M. Serral, M. Pinna, A. V. Zvelindovsky and J. B. Avalos, "Cell dynamics
simulations of sphere-forming diblock copolymers in thin films on chemically patterned
substrates," Macromolecules, vol. 49, pp. 1079-1092, 2016.
[73] J. D. Hoffman, Numerical Methods for Engineers and Scientist. United States:
McGRAW-HILL, 1992.
[74] S. Sirca and M. Horvat, Computational Methods for Physicists. Springer, 2010.
149
[75] S. P. Thampi, S. Ansumali, R. Adhikari and S. Succi, "Isotropic Discrete Laplacain
operators from lattice hydrodynamics," Journal of Computational Physics, vol. 234, pp.
1-7, 2012.
[76] S. A and O. Y, "Spinodal decomposition in 3-space," Physical Review E, vol. 48,
pp. 2622-2654, 1993.
[77] R. Dessí, M. Pinna and A. V. Zvelindovsky, "Cell dynamics simulations of
cylinder-forming diblock copolymers in thin films on topographical and chemically
patterned substrates," Macromolecules, vol. 46, pp. 1923-1931, 2013.
[78] T. Ohta and K. Kawasaki, "Equilibrium morphology of block copolymer melts,"
Macromolecules, vol. 19, pp. 2661-2632, 1986.
[79] L. Ludwik, "Theory of Microphase Separation in Block Copolymers,"
Macromolecules, vol. 13, pp. 1602-1617, 1980.
[80] J. Feng and E. Ruckenstein. Long-range ordered structures in diblock copolymer
melts induced by combined external fields. J. Chem. Phys. 121(3), pp. 1609-1625. 2004.
. DOI: http://dx.doi.org/10.1063/1.1763140.
[81] M. Patra and M. Karttunen, "Stencils with isotropic discretization error for
differential operators," Numerical Methods for Partial Differential Equations, vol. 22,
pp. 936-953, 2006.
[82] "Clipart - Blue grid sphere", Openclipart.org, 2016. [Online]. Available:
https://openclipart.org/detail/183675/blue-grid-sphere. [Accessed: 11- Apr-
2016].
150
[83]"Create_grid_with_VTK"[online]available:http://www.corephysics.com/wiki/index.
php [accessed: 11-Apr- 2016]
[84] X. Zhang, B. C. Berry, K. G. Yager, S. Kim, R. L. Jones, S. Satija, D. L. Pickel, J.
F. Douglas and A. Karim, "Surface morphology diagram for cylinder-forming block
copolymer thin films," ACSNANO, vol. 2, pp. 2331-2341, 2008.
[85] L. Li, K. Matsunaga, J. Zhu, T. Higuchi, H. Yabu, M. Shimomura, H. Jinnai, R. C.
Hayward and T. P. Russell, "Solvent-driven evolution of block copolymer morphology
under 3D confinement," Macromolecules, vol. 43, pp. 7807-7812, 2010.
[86] H. Xiang, K. Shin, T. Kim, S. Moon, T. McCarthy and T. Russell, "The influence
of confinement and curvature on the morphology of block copolymers," Journal of
Polymer Science Part B: Polymer Physics, vol. 43, pp. 3377-3383, 2005.
[87] X. Hongqi, s. Kyusoon, K. Taehyung, I. Sung Moon, T. J. McCarthy and T. P.
Russell, "- Block Copolymers under Cylindrical Confinement," - Macromolecules, vol.
37, pp. 5660-5664, 2004.
[88] D. Chen, S. Park, J. Chen, E. Redston and T. P. Russell, "A simple route for the
preparation of mesoporous nanostructures using block copolymers," ACSNANO, vol. 3,
pp. 2827-2833, 2009.
[89] J. M. Shin, M. P. Kim, H. Yang, K. H. Ku, S. G. Jang, K. H. Youm, G. Yi and B. J.
Kim, "Monodipserse Nanostructured Spheres of Block Copolymers and Nanoparticles
via Cross-Flow Membrane Emulsification," Chem. Mater., vol. 27, pp. 6314-6321,
09/22, 2015.
[90] M. J. Fasolka and A. M. Mayes, "BLOCK COPOLYMER THIN FILMS: Physics
and Applications1," Ann. Rev. Mat. Res., vol. 31, pp. 323-355, 2001.
151
[91] B. Yu, L. Baohui, Q. Jin, D. Ding and A. Shi, "Self-Assembly of Symmetric
Diblock Copolymers Confined in Spherical Nanopores," Marcomolecules, vol. 40, pp.
9133-9142, 2007.
[92] K. S. Lyakhova, G. J. A. Sevink, A. V. Zvelindovsky, A. Horvat and R. Magerle,
"Role of dissimlar interfaces in thin films of cylinder-forming block copolymers," J.
Chem. Phys., vol. 120, pp. 1127-1137, 2004.
[93] A. Horvat, K. S. Lyakhova, G. J. A. Sevink, A. V. Zvelindovsky and R. Magerle,
"Phase behaviour in thin films of cylinder-forming ABA block copolymers: Mesoscale
modelling," J. Chem. Phys., vol. 120, pp. 1117-1126, 2004.